backup 04_physworld 2026-11-08 10:11:26 (part 11)

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physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/INSTALLER

@@ -0,0 +1 @@
+pip

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/METADATA

@@ -0,0 +1,139 @@
+Metadata-Version: 2.4
+Name: numpy
+Version: 2.4.6
+Summary: Fundamental package for array computing in Python
+Author: Travis E. Oliphant et al.
+Maintainer-Email: NumPy Developers <numpy-discussion@python.org>
+License-Expression: BSD-3-Clause AND 0BSD AND MIT AND Zlib AND CC0-1.0
+License-File: LICENSE.txt
+License-File: numpy/_core/include/numpy/libdivide/LICENSE.txt
+License-File: numpy/_core/src/common/pythoncapi-compat/COPYING
+License-File: numpy/_core/src/highway/LICENSE
+License-File: numpy/_core/src/multiarray/dragon4_LICENSE.txt
+License-File: numpy/_core/src/npysort/x86-simd-sort/LICENSE.md
+License-File: numpy/_core/src/umath/svml/LICENSE
+License-File: numpy/fft/pocketfft/LICENSE.md
+License-File: numpy/linalg/lapack_lite/LICENSE.txt
+License-File: numpy/ma/LICENSE
+License-File: numpy/random/LICENSE.md
+License-File: numpy/random/src/distributions/LICENSE.md
+License-File: numpy/random/src/mt19937/LICENSE.md
+License-File: numpy/random/src/pcg64/LICENSE.md
+License-File: numpy/random/src/philox/LICENSE.md
+License-File: numpy/random/src/sfc64/LICENSE.md
+License-File: numpy/random/src/splitmix64/LICENSE.md
+Classifier: Development Status :: 5 - Production/Stable
+Classifier: Intended Audience :: Science/Research
+Classifier: Intended Audience :: Developers
+Classifier: Programming Language :: C
+Classifier: Programming Language :: Python
+Classifier: Programming Language :: Python :: 3
+Classifier: Programming Language :: Python :: 3.11
+Classifier: Programming Language :: Python :: 3.12
+Classifier: Programming Language :: Python :: 3.13
+Classifier: Programming Language :: Python :: 3.14
+Classifier: Programming Language :: Python :: 3 :: Only
+Classifier: Programming Language :: Python :: Implementation :: CPython
+Classifier: Topic :: Software Development
+Classifier: Topic :: Scientific/Engineering
+Classifier: Typing :: Typed
+Classifier: Operating System :: Microsoft :: Windows
+Classifier: Operating System :: POSIX
+Classifier: Operating System :: Unix
+Classifier: Operating System :: MacOS
+Project-URL: homepage, https://numpy.org
+Project-URL: documentation, https://numpy.org/doc/
+Project-URL: source, https://github.com/numpy/numpy
+Project-URL: download, https://pypi.org/project/numpy/#files
+Project-URL: tracker, https://github.com/numpy/numpy/issues
+Project-URL: release notes, https://numpy.org/doc/stable/release
+Requires-Python: >=3.11
+Description-Content-Type: text/markdown
+
+<h1 align="center">
+<img src="https://raw.githubusercontent.com/numpy/numpy/main/branding/logo/primary/numpylogo.svg" width="300">
+</h1><br>
+
+
+[![Powered by NumFOCUS](https://img.shields.io/badge/powered%20by-NumFOCUS-orange.svg?style=flat&colorA=E1523D&colorB=007D8A)](
+https://numfocus.org)
+[![PyPI Downloads](https://img.shields.io/pypi/dm/numpy.svg?label=PyPI%20downloads)](
+https://pypi.org/project/numpy/)
+[![Conda Downloads](https://img.shields.io/conda/dn/conda-forge/numpy.svg?label=Conda%20downloads)](
+https://anaconda.org/conda-forge/numpy)
+[![Stack Overflow](https://img.shields.io/badge/stackoverflow-Ask%20questions-blue.svg)](
+https://stackoverflow.com/questions/tagged/numpy)
+[![Nature Paper](https://img.shields.io/badge/DOI-10.1038%2Fs41586--020--2649--2-blue)](
+https://doi.org/10.1038/s41586-020-2649-2)
+[![LFX Health Score](https://insights.linuxfoundation.org/api/badge/health-score?project=numpy)](https://insights.linuxfoundation.org/project/numpy)
+[![OpenSSF Scorecard](https://api.securityscorecards.dev/projects/github.com/numpy/numpy/badge)](https://securityscorecards.dev/viewer/?uri=github.com/numpy/numpy)
+[![Typing](https://img.shields.io/pypi/types/numpy)](https://pypi.org/project/numpy/)
+
+
+NumPy is the fundamental package for scientific computing with Python.
+
+- **Website:** https://numpy.org
+- **Documentation:** https://numpy.org/doc
+- **Mailing list:** https://mail.python.org/mailman/listinfo/numpy-discussion
+- **Source code:** https://github.com/numpy/numpy
+- **Contributing:** https://numpy.org/devdocs/dev/index.html
+- **Bug reports:** https://github.com/numpy/numpy/issues
+- **Report a security vulnerability:** https://tidelift.com/docs/security
+
+It provides:
+
+- a powerful N-dimensional array object
+- sophisticated (broadcasting) functions
+- tools for integrating C/C++ and Fortran code
+- useful linear algebra, Fourier transform, and random number capabilities
+
+Testing:
+
+NumPy requires `pytest` and `hypothesis`.  Tests can then be run after installation with:
+
+    python -c "import numpy, sys; sys.exit(numpy.test() is False)"
+
+Code of Conduct
+----------------------
+
+NumPy is a community-driven open source project developed by a diverse group of
+[contributors](https://numpy.org/teams/). The NumPy leadership has made a strong
+commitment to creating an open, inclusive, and positive community. Please read the
+[NumPy Code of Conduct](https://numpy.org/code-of-conduct/) for guidance on how to interact
+with others in a way that makes our community thrive.
+
+Call for Contributions
+----------------------
+
+The NumPy project welcomes your expertise and enthusiasm!
+
+Small improvements or fixes are always appreciated. If you are considering larger contributions
+to the source code, please contact us through the [mailing
+list](https://mail.python.org/mailman/listinfo/numpy-discussion) first.
+
+Writing code isn’t the only way to contribute to NumPy. You can also:
+- review pull requests
+- help us stay on top of new and old issues
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+- maintain and improve [our website](https://github.com/numpy/numpy.org)
+- develop graphic design for our brand assets and promotional materials
+- translate website content
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+For more information about the ways you can contribute to NumPy, visit [our website](https://numpy.org/contribute/). 
+If you’re unsure where to start or how your skills fit in, reach out! You can
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+
+Our preferred channels of communication are all public, but if you’d like to
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+
+We also have a biweekly community call, details of which are announced on the
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+
+If you are new to contributing to open source, [this
+guide](https://opensource.guide/how-to-contribute/) helps explain why, what,
+and how to successfully get involved.

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/WHEEL

@@ -0,0 +1,6 @@
+Wheel-Version: 1.0
+Generator: meson
+Root-Is-Purelib: false
+Tag: cp311-cp311-manylinux_2_27_x86_64
+Tag: cp311-cp311-manylinux_2_28_x86_64
+

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/entry_points.txt

@@ -0,0 +1,13 @@
+[pkg_config]
+numpy = numpy._core.lib.pkgconfig
+
+[array_api]
+numpy = numpy
+
+[pyinstaller40]
+hook-dirs = numpy:_pyinstaller_hooks_dir
+
+[console_scripts]
+f2py = numpy.f2py.f2py2e:main
+numpy-config = numpy._configtool:main
+

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/licenses/LICENSE.txt

@@ -0,0 +1,935 @@
+Copyright (c) 2005-2025, NumPy Developers.
+All rights reserved.
+
+Redistribution and use in source and binary forms, with or without
+modification, are permitted provided that the following conditions are
+met:
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+
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+       contributors may be used to endorse or promote products derived
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+THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
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+OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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+DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
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+
+----
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+----
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+This binary distribution of NumPy also bundles the following software:
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+Files: numpy.libs/libscipy_openblas*.so
+Description: bundled as a dynamically linked library
+Availability: https://github.com/OpenMathLib/OpenBLAS/
+License: BSD-3-Clause
+  Copyright (c) 2011-2014, The OpenBLAS Project
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+  Redistribution and use in source and binary forms, with or without
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+Files: numpy.libs/libscipy_openblas*.so
+Description: bundled in OpenBLAS
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+License: BSD-3-Clause-Open-MPI
+  Copyright (c) 1992-2013 The University of Tennessee and The University
+                          of Tennessee Research Foundation.  All rights
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+Full text of license texts referred to above follows (that they are
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+----
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+GCC RUNTIME LIBRARY EXCEPTION
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+
+----
+
+                    GNU GENERAL PUBLIC LICENSE
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physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/licenses/numpy/_core/include/numpy/libdivide/LICENSE.txt

@@ -0,0 +1,21 @@
+  zlib License
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+  Copyright (C) 2016 - 2019 Kim Walisch, <kim.walisch@gmail.com>
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physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/licenses/numpy/_core/src/common/pythoncapi-compat/COPYING

@@ -0,0 +1,14 @@
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physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/licenses/numpy/_core/src/highway/LICENSE

@@ -0,0 +1,371 @@
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physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/licenses/numpy/_core/src/multiarray/dragon4_LICENSE.txt

@@ -0,0 +1,27 @@
+Copyright (c) 2014 Ryan Juckett
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+which has been ported from C++ to C and which has
+modifications specific to printing floats in numpy.
+
+Ryan Juckett's original code was under the Zlib license; he gave numpy
+permission to include it under the MIT license instead.

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/licenses/numpy/_core/src/npysort/x86-simd-sort/LICENSE.md

@@ -0,0 +1,28 @@
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+   this software without specific prior written permission.
+
+THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
+AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
+DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
+FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
+SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
+CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
+OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
+OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/licenses/numpy/_core/src/umath/svml/LICENSE

@@ -0,0 +1,30 @@
+Copyright (c) 2005-2021, NumPy Developers.
+All rights reserved.
+
+Redistribution and use in source and binary forms, with or without
+modification, are permitted provided that the following conditions are
+met:
+
+    * Redistributions of source code must retain the above copyright
+       notice, this list of conditions and the following disclaimer.
+
+    * Redistributions in binary form must reproduce the above
+       copyright notice, this list of conditions and the following
+       disclaimer in the documentation and/or other materials provided
+       with the distribution.
+
+    * Neither the name of the NumPy Developers nor the names of any
+       contributors may be used to endorse or promote products derived
+       from this software without specific prior written permission.
+
+THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
+LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
+A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
+OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
+LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
+DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
+THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
+(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
+OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/licenses/numpy/fft/pocketfft/LICENSE.md

@@ -0,0 +1,25 @@
+Copyright (C) 2010-2018 Max-Planck-Society
+All rights reserved.
+
+Redistribution and use in source and binary forms, with or without modification,
+are permitted provided that the following conditions are met:
+
+* Redistributions of source code must retain the above copyright notice, this
+  list of conditions and the following disclaimer.
+* Redistributions in binary form must reproduce the above copyright notice, this
+  list of conditions and the following disclaimer in the documentation and/or
+  other materials provided with the distribution.
+* Neither the name of the copyright holder nor the names of its contributors may
+  be used to endorse or promote products derived from this software without
+  specific prior written permission.
+
+THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
+ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
+WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
+DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR
+ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
+(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
+LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON
+ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
+(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/licenses/numpy/linalg/lapack_lite/LICENSE.txt

@@ -0,0 +1,48 @@
+Copyright (c) 1992-2013 The University of Tennessee and The University
+                        of Tennessee Research Foundation.  All rights
+                        reserved.
+Copyright (c) 2000-2013 The University of California Berkeley. All
+                        rights reserved.
+Copyright (c) 2006-2013 The University of Colorado Denver.  All rights
+                        reserved.
+
+$COPYRIGHT$
+
+Additional copyrights may follow
+
+$HEADER$
+
+Redistribution and use in source and binary forms, with or without
+modification, are permitted provided that the following conditions are
+met:
+
+- Redistributions of source code must retain the above copyright
+  notice, this list of conditions and the following disclaimer.
+
+- Redistributions in binary form must reproduce the above copyright
+  notice, this list of conditions and the following disclaimer listed
+  in this license in the documentation and/or other materials
+  provided with the distribution.
+
+- Neither the name of the copyright holders nor the names of its
+  contributors may be used to endorse or promote products derived from
+  this software without specific prior written permission.
+
+The copyright holders provide no reassurances that the source code
+provided does not infringe any patent, copyright, or any other
+intellectual property rights of third parties.  The copyright holders
+disclaim any liability to any recipient for claims brought against
+recipient by any third party for infringement of that parties
+intellectual property rights.
+
+THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
+LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
+A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
+OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
+LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
+DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
+THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
+(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
+OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/licenses/numpy/ma/LICENSE

@@ -0,0 +1,24 @@
+* Copyright (c) 2006, University of Georgia and Pierre G.F. Gerard-Marchant
+* All rights reserved.
+* Redistribution and use in source and binary forms, with or without
+* modification, are permitted provided that the following conditions are met:
+*
+*     * Redistributions of source code must retain the above copyright
+*       notice, this list of conditions and the following disclaimer.
+*     * Redistributions in binary form must reproduce the above copyright
+*       notice, this list of conditions and the following disclaimer in the
+*       documentation and/or other materials provided with the distribution.
+*     * Neither the name of the University of Georgia nor the
+*       names of its contributors may be used to endorse or promote products
+*       derived from this software without specific prior written permission.
+*
+* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND ANY
+* EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
+* WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
+* DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE FOR ANY
+* DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
+* (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
+* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
+* ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
+* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+* SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/licenses/numpy/random/LICENSE.md

@@ -0,0 +1,71 @@
+**This software is dual-licensed under the The University of Illinois/NCSA
+Open Source License (NCSA) and The 3-Clause BSD License**
+
+# NCSA Open Source License
+**Copyright (c) 2019 Kevin Sheppard. All rights reserved.**
+
+Developed by: Kevin Sheppard (<kevin.sheppard@economics.ox.ac.uk>,
+<kevin.k.sheppard@gmail.com>)
+[http://www.kevinsheppard.com](http://www.kevinsheppard.com)
+
+Permission is hereby granted, free of charge, to any person obtaining a copy of
+this software and associated documentation files (the "Software"), to deal with
+the Software without restriction, including without limitation the rights to
+use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies
+of the Software, and to permit persons to whom the Software is furnished to do
+so, subject to the following conditions:
+
+Redistributions of source code must retain the above copyright notice, this
+list of conditions and the following disclaimers.
+
+Redistributions in binary form must reproduce the above copyright notice, this
+list of conditions and the following disclaimers in the documentation and/or
+other materials provided with the distribution.
+
+Neither the names of Kevin Sheppard, nor the names of any contributors may be
+used to endorse or promote products derived from this Software without specific
+prior written permission.
+
+**THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS WITH
+THE SOFTWARE.**
+
+
+# 3-Clause BSD License
+**Copyright (c) 2019 Kevin Sheppard. All rights reserved.**
+
+Redistribution and use in source and binary forms, with or without
+modification, are permitted provided that the following conditions are met:
+
+1. Redistributions of source code must retain the above copyright notice,
+   this list of conditions and the following disclaimer.
+
+2. Redistributions in binary form must reproduce the above copyright notice,
+   this list of conditions and the following disclaimer in the documentation
+   and/or other materials provided with the distribution.
+
+3. Neither the name of the copyright holder nor the names of its contributors
+   may be used to endorse or promote products derived from this software
+   without specific prior written permission.
+
+**THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
+AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
+LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
+SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
+CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
+ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF
+THE POSSIBILITY OF SUCH DAMAGE.**
+
+# Components
+
+Many parts of this module have been derived from original sources, 
+often the algorithm's designer. Component licenses are located with 
+the component code.

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/licenses/numpy/random/src/distributions/LICENSE.md

@@ -0,0 +1,61 @@
+## NumPy
+
+Copyright (c) 2005-2017, NumPy Developers.
+All rights reserved.
+
+Redistribution and use in source and binary forms, with or without
+modification, are permitted provided that the following conditions are
+met:
+
+* Redistributions of source code must retain the above copyright
+   notice, this list of conditions and the following disclaimer.
+
+* Redistributions in binary form must reproduce the above
+   copyright notice, this list of conditions and the following
+   disclaimer in the documentation and/or other materials provided
+   with the distribution.
+
+* Neither the name of the NumPy Developers nor the names of any
+   contributors may be used to endorse or promote products derived
+   from this software without specific prior written permission.
+
+THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
+LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
+A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
+OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
+LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
+DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
+THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
+(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
+OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+
+
+## Julia
+
+The ziggurat methods were derived from Julia.
+
+Copyright (c) 2009-2019: Jeff Bezanson, Stefan Karpinski, Viral B. Shah,
+and other contributors:
+
+https://github.com/JuliaLang/julia/contributors
+
+Permission is hereby granted, free of charge, to any person obtaining
+a copy of this software and associated documentation files (the
+"Software"), to deal in the Software without restriction, including
+without limitation the rights to use, copy, modify, merge, publish,
+distribute, sublicense, and/or sell copies of the Software, and to
+permit persons to whom the Software is furnished to do so, subject to
+the following conditions:
+
+The above copyright notice and this permission notice shall be
+included in all copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
+MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
+NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE
+LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
+OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
+WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/licenses/numpy/random/src/mt19937/LICENSE.md

@@ -0,0 +1,61 @@
+# MT19937
+
+Copyright (c) 2003-2005, Jean-Sebastien Roy (js@jeannot.org)
+
+The rk_random and rk_seed functions algorithms and the original design of
+the Mersenne Twister RNG:
+
+  Copyright (C) 1997 - 2002, Makoto Matsumoto and Takuji Nishimura,
+  All rights reserved.
+
+  Redistribution and use in source and binary forms, with or without
+  modification, are permitted provided that the following conditions
+  are met:
+
+  1. Redistributions of source code must retain the above copyright
+  notice, this list of conditions and the following disclaimer.
+
+  2. Redistributions in binary form must reproduce the above copyright
+  notice, this list of conditions and the following disclaimer in the
+  documentation and/or other materials provided with the distribution.
+
+  3. The names of its contributors may not be used to endorse or promote
+  products derived from this software without specific prior written
+  permission.
+
+THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
+LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
+A PARTICULAR PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE COPYRIGHT OWNER
+OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
+EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
+PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
+PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
+LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+
+Original algorithm for the implementation of rk_interval function from
+Richard J. Wagner's implementation of the Mersenne Twister RNG, optimised by
+Magnus Jonsson.
+
+Constants used in the rk_double implementation by Isaku Wada.
+
+Permission is hereby granted, free of charge, to any person obtaining a
+copy of this software and associated documentation files (the
+"Software"), to deal in the Software without restriction, including
+without limitation the rights to use, copy, modify, merge, publish,
+distribute, sublicense, and/or sell copies of the Software, and to
+permit persons to whom the Software is furnished to do so, subject to
+the following conditions:
+
+The above copyright notice and this permission notice shall be included
+in all copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
+OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
+MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
+IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
+CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
+TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
+SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/licenses/numpy/random/src/pcg64/LICENSE.md

@@ -0,0 +1,22 @@
+# PCG64
+
+## The MIT License
+
+PCG Random Number Generation for C.
+
+Copyright 2014 Melissa O'Neill <oneill@pcg-random.org>
+
+Permission is hereby granted, free of charge, to any person obtaining 
+a copy of this software and associated documentation files (the "Software"), 
+to deal in the Software without restriction, including without limitation 
+the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in 
+all copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR 
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS 
+FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR 
+COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER 
+IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN 
+CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/licenses/numpy/random/src/philox/LICENSE.md

@@ -0,0 +1,31 @@
+# PHILOX
+
+Copyright 2010-2012, D. E. Shaw Research.
+All rights reserved.
+
+Redistribution and use in source and binary forms, with or without
+modification, are permitted provided that the following conditions are
+met:
+
+* Redistributions of source code must retain the above copyright
+  notice, this list of conditions, and the following disclaimer.
+
+* Redistributions in binary form must reproduce the above copyright
+  notice, this list of conditions, and the following disclaimer in the
+  documentation and/or other materials provided with the distribution.
+
+* Neither the name of D. E. Shaw Research nor the names of its
+  contributors may be used to endorse or promote products derived from
+  this software without specific prior written permission.
+
+THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
+LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
+A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
+OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
+LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
+DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
+THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
+(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
+OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/licenses/numpy/random/src/sfc64/LICENSE.md

@@ -0,0 +1,27 @@
+# SFC64
+
+## The MIT License
+
+Adapted from a C++ implementation of Chris Doty-Humphrey's SFC PRNG.
+
+https://gist.github.com/imneme/f1f7821f07cf76504a97f6537c818083
+
+Copyright (c) 2018 Melissa E. O'Neill
+
+Permission is hereby granted, free of charge, to any person obtaining a
+copy of this software and associated documentation files (the "Software"),
+to deal in the Software without restriction, including without limitation
+the rights to use, copy, modify, merge, publish, distribute, sublicense,
+and/or sell copies of the Software, and to permit persons to whom the
+Software is furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in
+all copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
+FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
+DEALINGS IN THE SOFTWARE.

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy-2.4.6.dist-info/licenses/numpy/random/src/splitmix64/LICENSE.md

@@ -0,0 +1,9 @@
+# SPLITMIX64
+
+Written in 2015 by Sebastiano Vigna (vigna@acm.org)
+
+To the extent possible under law, the author has dedicated all copyright
+and related and neighboring rights to this software to the public domain
+worldwide. This software is distributed without any warranty.
+
+See <http://creativecommons.org/publicdomain/zero/1.0/>.

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy.libs/libgfortran-040039e1-0352e75f.so.5.0.0

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:c6090048eccc763522c12ef016f81da6b627cb3a044f55cf0479a839c41c0980
+size 2833617

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy.libs/libquadmath-96973f99-934c22de.so.0.0.0

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:6ed5137f412781ad7863439fb543613f620b43c32b63292a0029246162f5bbc6
+size 250985

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy.libs/libscipy_openblas64_-32a4b2a6.so

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:05c9f9eb89ee68a4b9d673184fa91c99587e736392c0c2d49180a8aa5303d080
+size 25409073

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy/polynomial/_polytypes.pyi

@@ -0,0 +1,501 @@
+# ruff: noqa: PYI046
+
+from collections.abc import Sequence
+from typing import (
+    Any,
+    Literal,
+    NoReturn,
+    Protocol,
+    Self,
+    SupportsIndex,
+    SupportsInt,
+    TypeAlias,
+    TypeVar,
+    overload,
+    type_check_only,
+)
+
+import numpy as np
+import numpy.typing as npt
+from numpy._typing import (
+    _ArrayLikeComplex_co,
+    # array-likes
+    _ArrayLikeFloat_co,
+    _ArrayLikeNumber_co,
+    _ComplexLike_co,
+    _FloatLike_co,
+    # scalar-likes
+    _IntLike_co,
+    _NestedSequence,
+    _NumberLike_co,
+    _SupportsArray,
+)
+
+_T = TypeVar("_T")
+_T_contra = TypeVar("_T_contra", contravariant=True)
+_ScalarT = TypeVar("_ScalarT", bound=np.number | np.bool | np.object_)
+
+# compatible with e.g. int, float, complex, Decimal, Fraction, and ABCPolyBase
+@type_check_only
+class _SupportsCoefOps(Protocol[_T_contra]):
+    def __eq__(self, x: object, /) -> bool: ...
+    def __ne__(self, x: object, /) -> bool: ...
+    def __neg__(self, /) -> Self: ...
+    def __pos__(self, /) -> Self: ...
+    def __add__(self, x: _T_contra, /) -> Self: ...
+    def __sub__(self, x: _T_contra, /) -> Self: ...
+    def __mul__(self, x: _T_contra, /) -> Self: ...
+    def __pow__(self, x: _T_contra, /) -> Self | float: ...
+    def __radd__(self, x: _T_contra, /) -> Self: ...
+    def __rsub__(self, x: _T_contra, /) -> Self: ...
+    def __rmul__(self, x: _T_contra, /) -> Self: ...
+
+_Series: TypeAlias = np.ndarray[tuple[int], np.dtype[_ScalarT]]
+
+_FloatSeries: TypeAlias = _Series[np.floating]
+_ComplexSeries: TypeAlias = _Series[np.complexfloating]
+_ObjectSeries: TypeAlias = _Series[np.object_]
+_CoefSeries: TypeAlias = _Series[np.inexact | np.object_]
+
+_FloatArray: TypeAlias = npt.NDArray[np.floating]
+_ComplexArray: TypeAlias = npt.NDArray[np.complexfloating]
+_ObjectArray: TypeAlias = npt.NDArray[np.object_]
+_CoefArray: TypeAlias = npt.NDArray[np.inexact | np.object_]
+
+_Tuple2: TypeAlias = tuple[_T, _T]
+_Array1: TypeAlias = np.ndarray[tuple[Literal[1]], np.dtype[_ScalarT]]
+_Array2: TypeAlias = np.ndarray[tuple[Literal[2]], np.dtype[_ScalarT]]
+
+_AnyInt: TypeAlias = SupportsInt | SupportsIndex
+
+_CoefObjectLike_co: TypeAlias = np.object_ | _SupportsCoefOps[Any]
+_CoefLike_co: TypeAlias = _NumberLike_co | _CoefObjectLike_co
+
+# The term "series" is used here to refer to 1-d arrays of numeric scalars.
+_SeriesLikeBool_co: TypeAlias = _SupportsArray[np.dtype[np.bool]] | Sequence[bool | np.bool]
+_SeriesLikeInt_co: TypeAlias = _SupportsArray[np.dtype[np.integer | np.bool]] | Sequence[_IntLike_co]
+_SeriesLikeFloat_co: TypeAlias = _SupportsArray[np.dtype[np.floating | np.integer | np.bool]] | Sequence[_FloatLike_co]
+_SeriesLikeComplex_co: TypeAlias = _SupportsArray[np.dtype[np.number | np.bool]] | Sequence[_ComplexLike_co]
+_SeriesLikeObject_co: TypeAlias = _SupportsArray[np.dtype[np.object_]] | Sequence[_CoefObjectLike_co]
+_SeriesLikeCoef_co: TypeAlias = _SupportsArray[np.dtype[np.number | np.bool | np.object_]] | Sequence[_CoefLike_co]
+
+_ArrayLikeCoefObject_co: TypeAlias = _CoefObjectLike_co | _SeriesLikeObject_co | _NestedSequence[_SeriesLikeObject_co]
+_ArrayLikeCoef_co: TypeAlias = npt.NDArray[np.number | np.bool | np.object_] | _ArrayLikeNumber_co | _ArrayLikeCoefObject_co
+
+_Line: TypeAlias = np.ndarray[tuple[int], np.dtype[_ScalarT]]
+
+@type_check_only
+class _FuncLine(Protocol):
+    @overload
+    def __call__(self, /, off: _ScalarT, scl: _ScalarT) -> _Line[_ScalarT]: ...
+    @overload
+    def __call__(self, /, off: int, scl: int) -> _Line[np.int_]: ...
+    @overload
+    def __call__(self, /, off: float, scl: float) -> _Line[np.float64]: ...
+    @overload
+    def __call__(self, /, off: complex, scl: complex) -> _Line[np.complex128]: ...
+    @overload
+    def __call__(self, /, off: _SupportsCoefOps[Any], scl: _SupportsCoefOps[Any]) -> _Line[np.object_]: ...
+
+@type_check_only
+class _FuncFromRoots(Protocol):
+    @overload
+    def __call__(self, /, roots: _SeriesLikeFloat_co) -> _FloatSeries: ...
+    @overload
+    def __call__(self, /, roots: _SeriesLikeComplex_co) -> _ComplexSeries: ...
+    @overload
+    def __call__(self, /, roots: _SeriesLikeCoef_co) -> _ObjectSeries: ...
+
+@type_check_only
+class _FuncBinOp(Protocol):
+    @overload
+    def __call__(self, /, c1: _SeriesLikeBool_co, c2: _SeriesLikeBool_co) -> NoReturn: ...
+    @overload
+    def __call__(self, /, c1: _SeriesLikeFloat_co, c2: _SeriesLikeFloat_co) -> _FloatSeries: ...
+    @overload
+    def __call__(self, /, c1: _SeriesLikeComplex_co, c2: _SeriesLikeComplex_co) -> _ComplexSeries: ...
+    @overload
+    def __call__(self, /, c1: _SeriesLikeCoef_co, c2: _SeriesLikeCoef_co) -> _ObjectSeries: ...
+
+@type_check_only
+class _FuncUnOp(Protocol):
+    @overload
+    def __call__(self, /, c: _SeriesLikeFloat_co) -> _FloatSeries: ...
+    @overload
+    def __call__(self, /, c: _SeriesLikeComplex_co) -> _ComplexSeries: ...
+    @overload
+    def __call__(self, /, c: _SeriesLikeCoef_co) -> _ObjectSeries: ...
+
+@type_check_only
+class _FuncPoly2Ortho(Protocol):
+    @overload
+    def __call__(self, /, pol: _SeriesLikeFloat_co) -> _FloatSeries: ...
+    @overload
+    def __call__(self, /, pol: _SeriesLikeComplex_co) -> _ComplexSeries: ...
+    @overload
+    def __call__(self, /, pol: _SeriesLikeCoef_co) -> _ObjectSeries: ...
+
+@type_check_only
+class _FuncPow(Protocol):
+    @overload
+    def __call__(self, /, c: _SeriesLikeFloat_co, pow: _IntLike_co, maxpower: _IntLike_co | None = ...) -> _FloatSeries: ...
+    @overload
+    def __call__(self, /, c: _SeriesLikeComplex_co, pow: _IntLike_co, maxpower: _IntLike_co | None = ...) -> _ComplexSeries: ...
+    @overload
+    def __call__(self, /, c: _SeriesLikeCoef_co, pow: _IntLike_co, maxpower: _IntLike_co | None = ...) -> _ObjectSeries: ...
+
+@type_check_only
+class _FuncDer(Protocol):
+    @overload
+    def __call__(
+        self,
+        /,
+        c: _ArrayLikeFloat_co,
+        m: SupportsIndex = 1,
+        scl: _FloatLike_co = 1,
+        axis: SupportsIndex = 0,
+    ) -> _FloatArray: ...
+    @overload
+    def __call__(
+        self,
+        /,
+        c: _ArrayLikeComplex_co,
+        m: SupportsIndex = 1,
+        scl: _ComplexLike_co = 1,
+        axis: SupportsIndex = 0,
+    ) -> _ComplexArray: ...
+    @overload
+    def __call__(
+        self,
+        /,
+        c: _ArrayLikeCoef_co,
+        m: SupportsIndex = 1,
+        scl: _CoefLike_co = 1,
+        axis: SupportsIndex = 0,
+    ) -> _ObjectArray: ...
+
+@type_check_only
+class _FuncInteg(Protocol):
+    @overload
+    def __call__(
+        self,
+        /,
+        c: _ArrayLikeFloat_co,
+        m: SupportsIndex = 1,
+        k: _FloatLike_co | _SeriesLikeFloat_co = [],
+        lbnd: _FloatLike_co = 0,
+        scl: _FloatLike_co = 1,
+        axis: SupportsIndex = 0,
+    ) -> _FloatArray: ...
+    @overload
+    def __call__(
+        self,
+        /,
+        c: _ArrayLikeComplex_co,
+        m: SupportsIndex = 1,
+        k: _ComplexLike_co | _SeriesLikeComplex_co = [],
+        lbnd: _ComplexLike_co = 0,
+        scl: _ComplexLike_co = 1,
+        axis: SupportsIndex = 0,
+    ) -> _ComplexArray: ...
+    @overload
+    def __call__(
+        self,
+        /,
+        c: _ArrayLikeCoef_co,
+        m: SupportsIndex = 1,
+        k: _CoefLike_co | _SeriesLikeCoef_co = [],
+        lbnd: _CoefLike_co = 0,
+        scl: _CoefLike_co = 1,
+        axis: SupportsIndex = 0,
+    ) -> _ObjectArray: ...
+
+@type_check_only
+class _FuncVal(Protocol):
+    @overload
+    def __call__(self, /, x: _FloatLike_co, c: _SeriesLikeFloat_co, tensor: bool = True) -> np.floating: ...
+    @overload
+    def __call__(self, /, x: _NumberLike_co, c: _SeriesLikeComplex_co, tensor: bool = True) -> np.complexfloating: ...
+    @overload
+    def __call__(self, /, x: _ArrayLikeFloat_co, c: _ArrayLikeFloat_co, tensor: bool = True) -> _FloatArray: ...
+    @overload
+    def __call__(self, /, x: _ArrayLikeComplex_co, c: _ArrayLikeComplex_co, tensor: bool = True) -> _ComplexArray: ...
+    @overload
+    def __call__(self, /, x: _ArrayLikeCoef_co, c: _ArrayLikeCoef_co, tensor: bool = True) -> _ObjectArray: ...
+    @overload
+    def __call__(self, /, x: _CoefLike_co, c: _SeriesLikeObject_co, tensor: bool = True) -> _SupportsCoefOps[Any]: ...
+
+@type_check_only
+class _FuncVal2D(Protocol):
+    @overload
+    def __call__(self, /, x: _FloatLike_co, y: _FloatLike_co, c: _SeriesLikeFloat_co) -> np.floating: ...
+    @overload
+    def __call__(self, /, x: _NumberLike_co, y: _NumberLike_co, c: _SeriesLikeComplex_co) -> np.complexfloating: ...
+    @overload
+    def __call__(self, /, x: _ArrayLikeFloat_co, y: _ArrayLikeFloat_co, c: _ArrayLikeFloat_co) -> _FloatArray: ...
+    @overload
+    def __call__(self, /, x: _ArrayLikeComplex_co, y: _ArrayLikeComplex_co, c: _ArrayLikeComplex_co) -> _ComplexArray: ...
+    @overload
+    def __call__(self, /, x: _ArrayLikeCoef_co, y: _ArrayLikeCoef_co, c: _ArrayLikeCoef_co) -> _ObjectArray: ...
+    @overload
+    def __call__(self, /, x: _CoefLike_co, y: _CoefLike_co, c: _SeriesLikeCoef_co) -> _SupportsCoefOps[Any]: ...
+
+@type_check_only
+class _FuncVal3D(Protocol):
+    @overload
+    def __call__(
+        self,
+        /,
+        x: _FloatLike_co,
+        y: _FloatLike_co,
+        z: _FloatLike_co,
+        c: _SeriesLikeFloat_co,
+    ) -> np.floating: ...
+    @overload
+    def __call__(
+        self,
+        /,
+        x: _NumberLike_co,
+        y: _NumberLike_co,
+        z: _NumberLike_co,
+        c: _SeriesLikeComplex_co,
+    ) -> np.complexfloating: ...
+    @overload
+    def __call__(
+        self,
+        /,
+        x: _ArrayLikeFloat_co,
+        y: _ArrayLikeFloat_co,
+        z: _ArrayLikeFloat_co,
+        c: _ArrayLikeFloat_co,
+    ) -> _FloatArray: ...
+    @overload
+    def __call__(
+        self,
+        /,
+        x: _ArrayLikeComplex_co,
+        y: _ArrayLikeComplex_co,
+        z: _ArrayLikeComplex_co,
+        c: _ArrayLikeComplex_co,
+    ) -> _ComplexArray: ...
+    @overload
+    def __call__(
+        self,
+        /,
+        x: _ArrayLikeCoef_co,
+        y: _ArrayLikeCoef_co,
+        z: _ArrayLikeCoef_co,
+        c: _ArrayLikeCoef_co,
+    ) -> _ObjectArray: ...
+    @overload
+    def __call__(
+        self,
+        /,
+        x: _CoefLike_co,
+        y: _CoefLike_co,
+        z: _CoefLike_co,
+        c: _SeriesLikeCoef_co,
+    ) -> _SupportsCoefOps[Any]: ...
+
+@type_check_only
+class _FuncVander(Protocol):
+    @overload
+    def __call__(self, /, x: _ArrayLikeFloat_co, deg: SupportsIndex) -> _FloatArray: ...
+    @overload
+    def __call__(self, /, x: _ArrayLikeComplex_co, deg: SupportsIndex) -> _ComplexArray: ...
+    @overload
+    def __call__(self, /, x: _ArrayLikeCoef_co, deg: SupportsIndex) -> _ObjectArray: ...
+    @overload
+    def __call__(self, /, x: npt.ArrayLike, deg: SupportsIndex) -> _CoefArray: ...
+
+_AnyDegrees: TypeAlias = Sequence[SupportsIndex]
+
+@type_check_only
+class _FuncVander2D(Protocol):
+    @overload
+    def __call__(self, /, x: _ArrayLikeFloat_co, y: _ArrayLikeFloat_co, deg: _AnyDegrees) -> _FloatArray: ...
+    @overload
+    def __call__(self, /, x: _ArrayLikeComplex_co, y: _ArrayLikeComplex_co, deg: _AnyDegrees) -> _ComplexArray: ...
+    @overload
+    def __call__(self, /, x: _ArrayLikeCoef_co, y: _ArrayLikeCoef_co, deg: _AnyDegrees) -> _ObjectArray: ...
+    @overload
+    def __call__(self, /, x: npt.ArrayLike, y: npt.ArrayLike, deg: _AnyDegrees) -> _CoefArray: ...
+
+@type_check_only
+class _FuncVander3D(Protocol):
+    @overload
+    def __call__(
+        self,
+        /,
+        x: _ArrayLikeFloat_co,
+        y: _ArrayLikeFloat_co,
+        z: _ArrayLikeFloat_co,
+        deg: _AnyDegrees,
+    ) -> _FloatArray: ...
+    @overload
+    def __call__(
+        self,
+        /,
+        x: _ArrayLikeComplex_co,
+        y: _ArrayLikeComplex_co,
+        z: _ArrayLikeComplex_co,
+        deg: _AnyDegrees,
+    ) -> _ComplexArray: ...
+    @overload
+    def __call__(
+        self,
+        /,
+        x: _ArrayLikeCoef_co,
+        y: _ArrayLikeCoef_co,
+        z: _ArrayLikeCoef_co,
+        deg: _AnyDegrees,
+    ) -> _ObjectArray: ...
+    @overload
+    def __call__(
+        self,
+        /,
+        x: npt.ArrayLike,
+        y: npt.ArrayLike,
+        z: npt.ArrayLike,
+        deg: _AnyDegrees,
+    ) -> _CoefArray: ...
+
+_FullFitResult: TypeAlias = Sequence[np.inexact | np.int32]
+
+@type_check_only
+class _FuncFit(Protocol):
+    @overload
+    def __call__(
+        self,
+        /,
+        x: _SeriesLikeFloat_co,
+        y: _ArrayLikeFloat_co,
+        deg: int | _SeriesLikeInt_co,
+        rcond: float | None = ...,
+        full: Literal[False] = False,
+        w: _SeriesLikeFloat_co | None = ...,
+    ) -> _FloatArray: ...
+    @overload
+    def __call__(
+        self,
+        x: _SeriesLikeFloat_co,
+        y: _ArrayLikeFloat_co,
+        deg: int | _SeriesLikeInt_co,
+        rcond: float | None,
+        full: Literal[True],
+        /,
+        w: _SeriesLikeFloat_co | None = ...,
+    ) -> tuple[_FloatArray, _FullFitResult]: ...
+    @overload
+    def __call__(
+        self,
+        /,
+        x: _SeriesLikeFloat_co,
+        y: _ArrayLikeFloat_co,
+        deg: int | _SeriesLikeInt_co,
+        rcond: float | None = ...,
+        *,
+        full: Literal[True],
+        w: _SeriesLikeFloat_co | None = ...,
+    ) -> tuple[_FloatArray, _FullFitResult]: ...
+    @overload
+    def __call__(
+        self,
+        /,
+        x: _SeriesLikeComplex_co,
+        y: _ArrayLikeComplex_co,
+        deg: int | _SeriesLikeInt_co,
+        rcond: float | None = ...,
+        full: Literal[False] = False,
+        w: _SeriesLikeFloat_co | None = ...,
+    ) -> _ComplexArray: ...
+    @overload
+    def __call__(
+        self,
+        x: _SeriesLikeComplex_co,
+        y: _ArrayLikeComplex_co,
+        deg: int | _SeriesLikeInt_co,
+        rcond: float | None,
+        full: Literal[True],
+        /,
+        w: _SeriesLikeFloat_co | None = ...,
+    ) -> tuple[_ComplexArray, _FullFitResult]: ...
+    @overload
+    def __call__(
+        self,
+        /,
+        x: _SeriesLikeComplex_co,
+        y: _ArrayLikeComplex_co,
+        deg: int | _SeriesLikeInt_co,
+        rcond: float | None = ...,
+        *,
+        full: Literal[True],
+        w: _SeriesLikeFloat_co | None = ...,
+    ) -> tuple[_ComplexArray, _FullFitResult]: ...
+    @overload
+    def __call__(
+        self,
+        /,
+        x: _SeriesLikeComplex_co,
+        y: _ArrayLikeCoef_co,
+        deg: int | _SeriesLikeInt_co,
+        rcond: float | None = ...,
+        full: Literal[False] = False,
+        w: _SeriesLikeFloat_co | None = ...,
+    ) -> _ObjectArray: ...
+    @overload
+    def __call__(
+        self,
+        x: _SeriesLikeComplex_co,
+        y: _ArrayLikeCoef_co,
+        deg: int | _SeriesLikeInt_co,
+        rcond: float | None,
+        full: Literal[True],
+        /,
+        w: _SeriesLikeFloat_co | None = ...,
+    ) -> tuple[_ObjectArray, _FullFitResult]: ...
+    @overload
+    def __call__(
+        self,
+        /,
+        x: _SeriesLikeComplex_co,
+        y: _ArrayLikeCoef_co,
+        deg: int | _SeriesLikeInt_co,
+        rcond: float | None = ...,
+        *,
+        full: Literal[True],
+        w: _SeriesLikeFloat_co | None = ...,
+    ) -> tuple[_ObjectArray, _FullFitResult]: ...
+
+@type_check_only
+class _FuncRoots(Protocol):
+    @overload
+    def __call__(self, /, c: _SeriesLikeFloat_co) -> _Series[np.float64]: ...
+    @overload
+    def __call__(self, /, c: _SeriesLikeComplex_co) -> _Series[np.complex128]: ...
+    @overload
+    def __call__(self, /, c: _SeriesLikeCoef_co) -> _ObjectSeries: ...
+
+_Companion: TypeAlias = np.ndarray[tuple[int, int], np.dtype[_ScalarT]]
+
+@type_check_only
+class _FuncCompanion(Protocol):
+    @overload
+    def __call__(self, /, c: _SeriesLikeFloat_co) -> _Companion[np.float64]: ...
+    @overload
+    def __call__(self, /, c: _SeriesLikeComplex_co) -> _Companion[np.complex128]: ...
+    @overload
+    def __call__(self, /, c: _SeriesLikeCoef_co) -> _Companion[np.object_]: ...
+
+@type_check_only
+class _FuncGauss(Protocol):
+    def __call__(self, /, deg: SupportsIndex) -> _Tuple2[_Series[np.float64]]: ...
+
+@type_check_only
+class _FuncWeight(Protocol):
+    @overload
+    def __call__(self, /, x: _ArrayLikeFloat_co) -> npt.NDArray[np.float64]: ...
+    @overload
+    def __call__(self, /, x: _ArrayLikeComplex_co) -> npt.NDArray[np.complex128]: ...
+    @overload
+    def __call__(self, /, x: _ArrayLikeCoef_co) -> _ObjectArray: ...

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy/polynomial/chebyshev.py

@@ -0,0 +1,2001 @@
+"""
+====================================================
+Chebyshev Series (:mod:`numpy.polynomial.chebyshev`)
+====================================================
+
+This module provides a number of objects (mostly functions) useful for
+dealing with Chebyshev series, including a `Chebyshev` class that
+encapsulates the usual arithmetic operations.  (General information
+on how this module represents and works with such polynomials is in the
+docstring for its "parent" sub-package, `numpy.polynomial`).
+
+Classes
+-------
+
+.. autosummary::
+   :toctree: generated/
+
+   Chebyshev
+
+
+Constants
+---------
+
+.. autosummary::
+   :toctree: generated/
+
+   chebdomain
+   chebzero
+   chebone
+   chebx
+
+Arithmetic
+----------
+
+.. autosummary::
+   :toctree: generated/
+
+   chebadd
+   chebsub
+   chebmulx
+   chebmul
+   chebdiv
+   chebpow
+   chebval
+   chebval2d
+   chebval3d
+   chebgrid2d
+   chebgrid3d
+
+Calculus
+--------
+
+.. autosummary::
+   :toctree: generated/
+
+   chebder
+   chebint
+
+Misc Functions
+--------------
+
+.. autosummary::
+   :toctree: generated/
+
+   chebfromroots
+   chebroots
+   chebvander
+   chebvander2d
+   chebvander3d
+   chebgauss
+   chebweight
+   chebcompanion
+   chebfit
+   chebpts1
+   chebpts2
+   chebtrim
+   chebline
+   cheb2poly
+   poly2cheb
+   chebinterpolate
+
+See also
+--------
+`numpy.polynomial`
+
+Notes
+-----
+The implementations of multiplication, division, integration, and
+differentiation use the algebraic identities [1]_:
+
+.. math::
+    T_n(x) = \\frac{z^n + z^{-n}}{2} \\\\
+    z\\frac{dx}{dz} = \\frac{z - z^{-1}}{2}.
+
+where
+
+.. math:: x = \\frac{z + z^{-1}}{2}.
+
+These identities allow a Chebyshev series to be expressed as a finite,
+symmetric Laurent series.  In this module, this sort of Laurent series
+is referred to as a "z-series."
+
+References
+----------
+.. [1] A. T. Benjamin, et al., "Combinatorial Trigonometry with Chebyshev
+  Polynomials," *Journal of Statistical Planning and Inference 14*, 2008
+  (https://web.archive.org/web/20080221202153/https://www.math.hmc.edu/~benjamin/papers/CombTrig.pdf, pg. 4)
+
+"""  # noqa: E501
+import numpy as np
+
+from . import polyutils as pu
+from ._polybase import ABCPolyBase
+
+__all__ = [
+    'chebzero', 'chebone', 'chebx', 'chebdomain', 'chebline', 'chebadd',
+    'chebsub', 'chebmulx', 'chebmul', 'chebdiv', 'chebpow', 'chebval',
+    'chebder', 'chebint', 'cheb2poly', 'poly2cheb', 'chebfromroots',
+    'chebvander', 'chebfit', 'chebtrim', 'chebroots', 'chebpts1',
+    'chebpts2', 'Chebyshev', 'chebval2d', 'chebval3d', 'chebgrid2d',
+    'chebgrid3d', 'chebvander2d', 'chebvander3d', 'chebcompanion',
+    'chebgauss', 'chebweight', 'chebinterpolate']
+
+chebtrim = pu.trimcoef
+
+#
+# A collection of functions for manipulating z-series. These are private
+# functions and do minimal error checking.
+#
+
+def _cseries_to_zseries(c):
+    """Convert Chebyshev series to z-series.
+
+    Convert a Chebyshev series to the equivalent z-series. The result is
+    never an empty array. The dtype of the return is the same as that of
+    the input. No checks are run on the arguments as this routine is for
+    internal use.
+
+    Parameters
+    ----------
+    c : 1-D ndarray
+        Chebyshev coefficients, ordered from low to high
+
+    Returns
+    -------
+    zs : 1-D ndarray
+        Odd length symmetric z-series, ordered from  low to high.
+
+    """
+    n = c.size
+    zs = np.zeros(2 * n - 1, dtype=c.dtype)
+    zs[n - 1:] = c / 2
+    return zs + zs[::-1]
+
+
+def _zseries_to_cseries(zs):
+    """Convert z-series to a Chebyshev series.
+
+    Convert a z series to the equivalent Chebyshev series. The result is
+    never an empty array. The dtype of the return is the same as that of
+    the input. No checks are run on the arguments as this routine is for
+    internal use.
+
+    Parameters
+    ----------
+    zs : 1-D ndarray
+        Odd length symmetric z-series, ordered from  low to high.
+
+    Returns
+    -------
+    c : 1-D ndarray
+        Chebyshev coefficients, ordered from  low to high.
+
+    """
+    n = (zs.size + 1) // 2
+    c = zs[n - 1:].copy()
+    c[1:n] *= 2
+    return c
+
+
+def _zseries_mul(z1, z2):
+    """Multiply two z-series.
+
+    Multiply two z-series to produce a z-series.
+
+    Parameters
+    ----------
+    z1, z2 : 1-D ndarray
+        The arrays must be 1-D but this is not checked.
+
+    Returns
+    -------
+    product : 1-D ndarray
+        The product z-series.
+
+    Notes
+    -----
+    This is simply convolution. If symmetric/anti-symmetric z-series are
+    denoted by S/A then the following rules apply:
+
+    S*S, A*A -> S
+    S*A, A*S -> A
+
+    """
+    return np.convolve(z1, z2)
+
+
+def _zseries_div(z1, z2):
+    """Divide the first z-series by the second.
+
+    Divide `z1` by `z2` and return the quotient and remainder as z-series.
+    Warning: this implementation only applies when both z1 and z2 have the
+    same symmetry, which is sufficient for present purposes.
+
+    Parameters
+    ----------
+    z1, z2 : 1-D ndarray
+        The arrays must be 1-D and have the same symmetry, but this is not
+        checked.
+
+    Returns
+    -------
+
+    (quotient, remainder) : 1-D ndarrays
+        Quotient and remainder as z-series.
+
+    Notes
+    -----
+    This is not the same as polynomial division on account of the desired form
+    of the remainder. If symmetric/anti-symmetric z-series are denoted by S/A
+    then the following rules apply:
+
+    S/S -> S,S
+    A/A -> S,A
+
+    The restriction to types of the same symmetry could be fixed but seems like
+    unneeded generality. There is no natural form for the remainder in the case
+    where there is no symmetry.
+
+    """
+    z1 = z1.copy()
+    z2 = z2.copy()
+    lc1 = len(z1)
+    lc2 = len(z2)
+    if lc2 == 1:
+        z1 /= z2
+        return z1, z1[:1] * 0
+    elif lc1 < lc2:
+        return z1[:1] * 0, z1
+    else:
+        dlen = lc1 - lc2
+        scl = z2[0]
+        z2 /= scl
+        quo = np.empty(dlen + 1, dtype=z1.dtype)
+        i = 0
+        j = dlen
+        while i < j:
+            r = z1[i]
+            quo[i] = z1[i]
+            quo[dlen - i] = r
+            tmp = r * z2
+            z1[i:i + lc2] -= tmp
+            z1[j:j + lc2] -= tmp
+            i += 1
+            j -= 1
+        r = z1[i]
+        quo[i] = r
+        tmp = r * z2
+        z1[i:i + lc2] -= tmp
+        quo /= scl
+        rem = z1[i + 1:i - 1 + lc2].copy()
+        return quo, rem
+
+
+def _zseries_der(zs):
+    """Differentiate a z-series.
+
+    The derivative is with respect to x, not z. This is achieved using the
+    chain rule and the value of dx/dz given in the module notes.
+
+    Parameters
+    ----------
+    zs : z-series
+        The z-series to differentiate.
+
+    Returns
+    -------
+    derivative : z-series
+        The derivative
+
+    Notes
+    -----
+    The zseries for x (ns) has been multiplied by two in order to avoid
+    using floats that are incompatible with Decimal and likely other
+    specialized scalar types. This scaling has been compensated by
+    multiplying the value of zs by two also so that the two cancels in the
+    division.
+
+    """
+    n = len(zs) // 2
+    ns = np.array([-1, 0, 1], dtype=zs.dtype)
+    zs *= np.arange(-n, n + 1) * 2
+    d, r = _zseries_div(zs, ns)
+    return d
+
+
+def _zseries_int(zs):
+    """Integrate a z-series.
+
+    The integral is with respect to x, not z. This is achieved by a change
+    of variable using dx/dz given in the module notes.
+
+    Parameters
+    ----------
+    zs : z-series
+        The z-series to integrate
+
+    Returns
+    -------
+    integral : z-series
+        The indefinite integral
+
+    Notes
+    -----
+    The zseries for x (ns) has been multiplied by two in order to avoid
+    using floats that are incompatible with Decimal and likely other
+    specialized scalar types. This scaling has been compensated by
+    dividing the resulting zs by two.
+
+    """
+    n = 1 + len(zs) // 2
+    ns = np.array([-1, 0, 1], dtype=zs.dtype)
+    zs = _zseries_mul(zs, ns)
+    div = np.arange(-n, n + 1) * 2
+    zs[:n] /= div[:n]
+    zs[n + 1:] /= div[n + 1:]
+    zs[n] = 0
+    return zs
+
+#
+# Chebyshev series functions
+#
+
+
+def poly2cheb(pol):
+    """
+    Convert a polynomial to a Chebyshev series.
+
+    Convert an array representing the coefficients of a polynomial (relative
+    to the "standard" basis) ordered from lowest degree to highest, to an
+    array of the coefficients of the equivalent Chebyshev series, ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    pol : array_like
+        1-D array containing the polynomial coefficients
+
+    Returns
+    -------
+    c : ndarray
+        1-D array containing the coefficients of the equivalent Chebyshev
+        series.
+
+    See Also
+    --------
+    cheb2poly
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> from numpy import polynomial as P
+    >>> p = P.Polynomial(range(4))
+    >>> p
+    Polynomial([0., 1., 2., 3.], domain=[-1.,  1.], window=[-1.,  1.], symbol='x')
+    >>> c = p.convert(kind=P.Chebyshev)
+    >>> c
+    Chebyshev([1.  , 3.25, 1.  , 0.75], domain=[-1.,  1.], window=[-1., ...
+    >>> P.chebyshev.poly2cheb(range(4))
+    array([1.  , 3.25, 1.  , 0.75])
+
+    """
+    [pol] = pu.as_series([pol])
+    deg = len(pol) - 1
+    res = 0
+    for i in range(deg, -1, -1):
+        res = chebadd(chebmulx(res), pol[i])
+    return res
+
+
+def cheb2poly(c):
+    """
+    Convert a Chebyshev series to a polynomial.
+
+    Convert an array representing the coefficients of a Chebyshev series,
+    ordered from lowest degree to highest, to an array of the coefficients
+    of the equivalent polynomial (relative to the "standard" basis) ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array containing the Chebyshev series coefficients, ordered
+        from lowest order term to highest.
+
+    Returns
+    -------
+    pol : ndarray
+        1-D array containing the coefficients of the equivalent polynomial
+        (relative to the "standard" basis) ordered from lowest order term
+        to highest.
+
+    See Also
+    --------
+    poly2cheb
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> from numpy import polynomial as P
+    >>> c = P.Chebyshev(range(4))
+    >>> c
+    Chebyshev([0., 1., 2., 3.], domain=[-1.,  1.], window=[-1.,  1.], symbol='x')
+    >>> p = c.convert(kind=P.Polynomial)
+    >>> p
+    Polynomial([-2., -8.,  4., 12.], domain=[-1.,  1.], window=[-1.,  1.], ...
+    >>> P.chebyshev.cheb2poly(range(4))
+    array([-2.,  -8.,   4.,  12.])
+
+    """
+    from .polynomial import polyadd, polymulx, polysub
+
+    [c] = pu.as_series([c])
+    n = len(c)
+    if n < 3:
+        return c
+    else:
+        c0 = c[-2]
+        c1 = c[-1]
+        # i is the current degree of c1
+        for i in range(n - 1, 1, -1):
+            tmp = c0
+            c0 = polysub(c[i - 2], c1)
+            c1 = polyadd(tmp, polymulx(c1) * 2)
+        return polyadd(c0, polymulx(c1))
+
+
+#
+# These are constant arrays are of integer type so as to be compatible
+# with the widest range of other types, such as Decimal.
+#
+
+# Chebyshev default domain.
+chebdomain = np.array([-1., 1.])
+
+# Chebyshev coefficients representing zero.
+chebzero = np.array([0])
+
+# Chebyshev coefficients representing one.
+chebone = np.array([1])
+
+# Chebyshev coefficients representing the identity x.
+chebx = np.array([0, 1])
+
+
+def chebline(off, scl):
+    """
+    Chebyshev series whose graph is a straight line.
+
+    Parameters
+    ----------
+    off, scl : scalars
+        The specified line is given by ``off + scl*x``.
+
+    Returns
+    -------
+    y : ndarray
+        This module's representation of the Chebyshev series for
+        ``off + scl*x``.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyline
+    numpy.polynomial.legendre.legline
+    numpy.polynomial.laguerre.lagline
+    numpy.polynomial.hermite.hermline
+    numpy.polynomial.hermite_e.hermeline
+
+    Examples
+    --------
+    >>> import numpy.polynomial.chebyshev as C
+    >>> C.chebline(3,2)
+    array([3, 2])
+    >>> C.chebval(-3, C.chebline(3,2)) # should be -3
+    -3.0
+
+    """
+    if scl != 0:
+        return np.array([off, scl])
+    else:
+        return np.array([off])
+
+
+def chebfromroots(roots):
+    """
+    Generate a Chebyshev series with given roots.
+
+    The function returns the coefficients of the polynomial
+
+    .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
+
+    in Chebyshev form, where the :math:`r_n` are the roots specified in
+    `roots`.  If a zero has multiplicity n, then it must appear in `roots`
+    n times.  For instance, if 2 is a root of multiplicity three and 3 is a
+    root of multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3].
+    The roots can appear in any order.
+
+    If the returned coefficients are `c`, then
+
+    .. math:: p(x) = c_0 + c_1 * T_1(x) + ... +  c_n * T_n(x)
+
+    The coefficient of the last term is not generally 1 for monic
+    polynomials in Chebyshev form.
+
+    Parameters
+    ----------
+    roots : array_like
+        Sequence containing the roots.
+
+    Returns
+    -------
+    out : ndarray
+        1-D array of coefficients.  If all roots are real then `out` is a
+        real array, if some of the roots are complex, then `out` is complex
+        even if all the coefficients in the result are real (see Examples
+        below).
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyfromroots
+    numpy.polynomial.legendre.legfromroots
+    numpy.polynomial.laguerre.lagfromroots
+    numpy.polynomial.hermite.hermfromroots
+    numpy.polynomial.hermite_e.hermefromroots
+
+    Examples
+    --------
+    >>> import numpy.polynomial.chebyshev as C
+    >>> C.chebfromroots((-1,0,1)) # x^3 - x relative to the standard basis
+    array([ 0.  , -0.25,  0.  ,  0.25])
+    >>> j = complex(0,1)
+    >>> C.chebfromroots((-j,j)) # x^2 + 1 relative to the standard basis
+    array([1.5+0.j, 0. +0.j, 0.5+0.j])
+
+    """
+    return pu._fromroots(chebline, chebmul, roots)
+
+
+def chebadd(c1, c2):
+    """
+    Add one Chebyshev series to another.
+
+    Returns the sum of two Chebyshev series `c1` + `c2`.  The arguments
+    are sequences of coefficients ordered from lowest order term to
+    highest, i.e., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Chebyshev series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the Chebyshev series of their sum.
+
+    See Also
+    --------
+    chebsub, chebmulx, chebmul, chebdiv, chebpow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the sum of two Chebyshev series
+    is a Chebyshev series (without having to "reproject" the result onto
+    the basis set) so addition, just like that of "standard" polynomials,
+    is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial import chebyshev as C
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2,1)
+    >>> C.chebadd(c1,c2)
+    array([4., 4., 4.])
+
+    """
+    return pu._add(c1, c2)
+
+
+def chebsub(c1, c2):
+    """
+    Subtract one Chebyshev series from another.
+
+    Returns the difference of two Chebyshev series `c1` - `c2`.  The
+    sequences of coefficients are from lowest order term to highest, i.e.,
+    [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Chebyshev series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Chebyshev series coefficients representing their difference.
+
+    See Also
+    --------
+    chebadd, chebmulx, chebmul, chebdiv, chebpow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the difference of two Chebyshev
+    series is a Chebyshev series (without having to "reproject" the result
+    onto the basis set) so subtraction, just like that of "standard"
+    polynomials, is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial import chebyshev as C
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2,1)
+    >>> C.chebsub(c1,c2)
+    array([-2.,  0.,  2.])
+    >>> C.chebsub(c2,c1) # -C.chebsub(c1,c2)
+    array([ 2.,  0., -2.])
+
+    """
+    return pu._sub(c1, c2)
+
+
+def chebmulx(c):
+    """Multiply a Chebyshev series by x.
+
+    Multiply the polynomial `c` by x, where x is the independent
+    variable.
+
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Chebyshev series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the result of the multiplication.
+
+    See Also
+    --------
+    chebadd, chebsub, chebmul, chebdiv, chebpow
+
+    Examples
+    --------
+    >>> from numpy.polynomial import chebyshev as C
+    >>> C.chebmulx([1,2,3])
+    array([1. , 2.5, 1. , 1.5])
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    # The zero series needs special treatment
+    if len(c) == 1 and c[0] == 0:
+        return c
+
+    prd = np.empty(len(c) + 1, dtype=c.dtype)
+    prd[0] = c[0] * 0
+    prd[1] = c[0]
+    if len(c) > 1:
+        tmp = c[1:] / 2
+        prd[2:] = tmp
+        prd[0:-2] += tmp
+    return prd
+
+
+def chebmul(c1, c2):
+    """
+    Multiply one Chebyshev series by another.
+
+    Returns the product of two Chebyshev series `c1` * `c2`.  The arguments
+    are sequences of coefficients, from lowest order "term" to highest,
+    e.g., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Chebyshev series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Chebyshev series coefficients representing their product.
+
+    See Also
+    --------
+    chebadd, chebsub, chebmulx, chebdiv, chebpow
+
+    Notes
+    -----
+    In general, the (polynomial) product of two C-series results in terms
+    that are not in the Chebyshev polynomial basis set.  Thus, to express
+    the product as a C-series, it is typically necessary to "reproject"
+    the product onto said basis set, which typically produces
+    "unintuitive live" (but correct) results; see Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import chebyshev as C
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2,1)
+    >>> C.chebmul(c1,c2) # multiplication requires "reprojection"
+    array([  6.5,  12. ,  12. ,   4. ,   1.5])
+
+    """
+    # c1, c2 are trimmed copies
+    [c1, c2] = pu.as_series([c1, c2])
+    z1 = _cseries_to_zseries(c1)
+    z2 = _cseries_to_zseries(c2)
+    prd = _zseries_mul(z1, z2)
+    ret = _zseries_to_cseries(prd)
+    return pu.trimseq(ret)
+
+
+def chebdiv(c1, c2):
+    """
+    Divide one Chebyshev series by another.
+
+    Returns the quotient-with-remainder of two Chebyshev series
+    `c1` / `c2`.  The arguments are sequences of coefficients from lowest
+    order "term" to highest, e.g., [1,2,3] represents the series
+    ``T_0 + 2*T_1 + 3*T_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Chebyshev series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    [quo, rem] : ndarrays
+        Of Chebyshev series coefficients representing the quotient and
+        remainder.
+
+    See Also
+    --------
+    chebadd, chebsub, chebmulx, chebmul, chebpow
+
+    Notes
+    -----
+    In general, the (polynomial) division of one C-series by another
+    results in quotient and remainder terms that are not in the Chebyshev
+    polynomial basis set.  Thus, to express these results as C-series, it
+    is typically necessary to "reproject" the results onto said basis
+    set, which typically produces "unintuitive" (but correct) results;
+    see Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import chebyshev as C
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2,1)
+    >>> C.chebdiv(c1,c2) # quotient "intuitive," remainder not
+    (array([3.]), array([-8., -4.]))
+    >>> c2 = (0,1,2,3)
+    >>> C.chebdiv(c2,c1) # neither "intuitive"
+    (array([0., 2.]), array([-2., -4.]))
+
+    """
+    # c1, c2 are trimmed copies
+    [c1, c2] = pu.as_series([c1, c2])
+    if c2[-1] == 0:
+        raise ZeroDivisionError  # FIXME: add message with details to exception
+
+    # note: this is more efficient than `pu._div(chebmul, c1, c2)`
+    lc1 = len(c1)
+    lc2 = len(c2)
+    if lc1 < lc2:
+        return c1[:1] * 0, c1
+    elif lc2 == 1:
+        return c1 / c2[-1], c1[:1] * 0
+    else:
+        z1 = _cseries_to_zseries(c1)
+        z2 = _cseries_to_zseries(c2)
+        quo, rem = _zseries_div(z1, z2)
+        quo = pu.trimseq(_zseries_to_cseries(quo))
+        rem = pu.trimseq(_zseries_to_cseries(rem))
+        return quo, rem
+
+
+def chebpow(c, pow, maxpower=16):
+    """Raise a Chebyshev series to a power.
+
+    Returns the Chebyshev series `c` raised to the power `pow`. The
+    argument `c` is a sequence of coefficients ordered from low to high.
+    i.e., [1,2,3] is the series  ``T_0 + 2*T_1 + 3*T_2.``
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Chebyshev series coefficients ordered from low to
+        high.
+    pow : integer
+        Power to which the series will be raised
+    maxpower : integer, optional
+        Maximum power allowed. This is mainly to limit growth of the series
+        to unmanageable size. Default is 16
+
+    Returns
+    -------
+    coef : ndarray
+        Chebyshev series of power.
+
+    See Also
+    --------
+    chebadd, chebsub, chebmulx, chebmul, chebdiv
+
+    Examples
+    --------
+    >>> from numpy.polynomial import chebyshev as C
+    >>> C.chebpow([1, 2, 3, 4], 2)
+    array([15.5, 22. , 16. , ..., 12.5, 12. ,  8. ])
+
+    """
+    # note: this is more efficient than `pu._pow(chebmul, c1, c2)`, as it
+    # avoids converting between z and c series repeatedly
+
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    power = int(pow)
+    if power != pow or power < 0:
+        raise ValueError("Power must be a non-negative integer.")
+    elif maxpower is not None and power > maxpower:
+        raise ValueError("Power is too large")
+    elif power == 0:
+        return np.array([1], dtype=c.dtype)
+    elif power == 1:
+        return c
+    else:
+        # This can be made more efficient by using powers of two
+        # in the usual way.
+        zs = _cseries_to_zseries(c)
+        prd = zs
+        for i in range(2, power + 1):
+            prd = np.convolve(prd, zs)
+        return _zseries_to_cseries(prd)
+
+
+def chebder(c, m=1, scl=1, axis=0):
+    """
+    Differentiate a Chebyshev series.
+
+    Returns the Chebyshev series coefficients `c` differentiated `m` times
+    along `axis`.  At each iteration the result is multiplied by `scl` (the
+    scaling factor is for use in a linear change of variable). The argument
+    `c` is an array of coefficients from low to high degree along each
+    axis, e.g., [1,2,3] represents the series ``1*T_0 + 2*T_1 + 3*T_2``
+    while [[1,2],[1,2]] represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) +
+    2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is
+    ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Chebyshev series coefficients. If c is multidimensional
+        the different axis correspond to different variables with the
+        degree in each axis given by the corresponding index.
+    m : int, optional
+        Number of derivatives taken, must be non-negative. (Default: 1)
+    scl : scalar, optional
+        Each differentiation is multiplied by `scl`.  The end result is
+        multiplication by ``scl**m``.  This is for use in a linear change of
+        variable. (Default: 1)
+    axis : int, optional
+        Axis over which the derivative is taken. (Default: 0).
+
+    Returns
+    -------
+    der : ndarray
+        Chebyshev series of the derivative.
+
+    See Also
+    --------
+    chebint
+
+    Notes
+    -----
+    In general, the result of differentiating a C-series needs to be
+    "reprojected" onto the C-series basis set. Thus, typically, the
+    result of this function is "unintuitive," albeit correct; see Examples
+    section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import chebyshev as C
+    >>> c = (1,2,3,4)
+    >>> C.chebder(c)
+    array([14., 12., 24.])
+    >>> C.chebder(c,3)
+    array([96.])
+    >>> C.chebder(c,scl=-1)
+    array([-14., -12., -24.])
+    >>> C.chebder(c,2,-1)
+    array([12.,  96.])
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    cnt = pu._as_int(m, "the order of derivation")
+    iaxis = pu._as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of derivation must be non-negative")
+    iaxis = np.lib.array_utils.normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    n = len(c)
+    if cnt >= n:
+        c = c[:1] * 0
+    else:
+        for i in range(cnt):
+            n = n - 1
+            c *= scl
+            der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
+            for j in range(n, 2, -1):
+                der[j - 1] = (2 * j) * c[j]
+                c[j - 2] += (j * c[j]) / (j - 2)
+            if n > 1:
+                der[1] = 4 * c[2]
+            der[0] = c[1]
+            c = der
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def chebint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
+    """
+    Integrate a Chebyshev series.
+
+    Returns the Chebyshev series coefficients `c` integrated `m` times from
+    `lbnd` along `axis`. At each iteration the resulting series is
+    **multiplied** by `scl` and an integration constant, `k`, is added.
+    The scaling factor is for use in a linear change of variable.  ("Buyer
+    beware": note that, depending on what one is doing, one may want `scl`
+    to be the reciprocal of what one might expect; for more information,
+    see the Notes section below.)  The argument `c` is an array of
+    coefficients from low to high degree along each axis, e.g., [1,2,3]
+    represents the series ``T_0 + 2*T_1 + 3*T_2`` while [[1,2],[1,2]]
+    represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) +
+    2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Chebyshev series coefficients. If c is multidimensional
+        the different axis correspond to different variables with the
+        degree in each axis given by the corresponding index.
+    m : int, optional
+        Order of integration, must be positive. (Default: 1)
+    k : {[], list, scalar}, optional
+        Integration constant(s).  The value of the first integral at zero
+        is the first value in the list, the value of the second integral
+        at zero is the second value, etc.  If ``k == []`` (the default),
+        all constants are set to zero.  If ``m == 1``, a single scalar can
+        be given instead of a list.
+    lbnd : scalar, optional
+        The lower bound of the integral. (Default: 0)
+    scl : scalar, optional
+        Following each integration the result is *multiplied* by `scl`
+        before the integration constant is added. (Default: 1)
+    axis : int, optional
+        Axis over which the integral is taken. (Default: 0).
+
+    Returns
+    -------
+    S : ndarray
+        C-series coefficients of the integral.
+
+    Raises
+    ------
+    ValueError
+        If ``m < 1``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
+        ``np.ndim(scl) != 0``.
+
+    See Also
+    --------
+    chebder
+
+    Notes
+    -----
+    Note that the result of each integration is *multiplied* by `scl`.
+    Why is this important to note?  Say one is making a linear change of
+    variable :math:`u = ax + b` in an integral relative to `x`.  Then
+    :math:`dx = du/a`, so one will need to set `scl` equal to
+    :math:`1/a`- perhaps not what one would have first thought.
+
+    Also note that, in general, the result of integrating a C-series needs
+    to be "reprojected" onto the C-series basis set.  Thus, typically,
+    the result of this function is "unintuitive," albeit correct; see
+    Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import chebyshev as C
+    >>> c = (1,2,3)
+    >>> C.chebint(c)
+    array([ 0.5, -0.5,  0.5,  0.5])
+    >>> C.chebint(c,3)
+    array([ 0.03125   , -0.1875    ,  0.04166667, -0.05208333,  0.01041667, # may vary
+        0.00625   ])
+    >>> C.chebint(c, k=3)
+    array([ 3.5, -0.5,  0.5,  0.5])
+    >>> C.chebint(c,lbnd=-2)
+    array([ 8.5, -0.5,  0.5,  0.5])
+    >>> C.chebint(c,scl=-2)
+    array([-1.,  1., -1., -1.])
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if not np.iterable(k):
+        k = [k]
+    cnt = pu._as_int(m, "the order of integration")
+    iaxis = pu._as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of integration must be non-negative")
+    if len(k) > cnt:
+        raise ValueError("Too many integration constants")
+    if np.ndim(lbnd) != 0:
+        raise ValueError("lbnd must be a scalar.")
+    if np.ndim(scl) != 0:
+        raise ValueError("scl must be a scalar.")
+    iaxis = np.lib.array_utils.normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    k = list(k) + [0] * (cnt - len(k))
+    for i in range(cnt):
+        n = len(c)
+        c *= scl
+        if n == 1 and np.all(c[0] == 0):
+            c[0] += k[i]
+        else:
+            tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype)
+            tmp[0] = c[0] * 0
+            tmp[1] = c[0]
+            if n > 1:
+                tmp[2] = c[1] / 4
+            for j in range(2, n):
+                tmp[j + 1] = c[j] / (2 * (j + 1))
+                tmp[j - 1] -= c[j] / (2 * (j - 1))
+            tmp[0] += k[i] - chebval(lbnd, tmp)
+            c = tmp
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def chebval(x, c, tensor=True):
+    """
+    Evaluate a Chebyshev series at points x.
+
+    If `c` is of length `n + 1`, this function returns the value:
+
+    .. math:: p(x) = c_0 * T_0(x) + c_1 * T_1(x) + ... + c_n * T_n(x)
+
+    The parameter `x` is converted to an array only if it is a tuple or a
+    list, otherwise it is treated as a scalar. In either case, either `x`
+    or its elements must support multiplication and addition both with
+    themselves and with the elements of `c`.
+
+    If `c` is a 1-D array, then ``p(x)`` will have the same shape as `x`.  If
+    `c` is multidimensional, then the shape of the result depends on the
+    value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
+    x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
+    scalars have shape (,).
+
+    Trailing zeros in the coefficients will be used in the evaluation, so
+    they should be avoided if efficiency is a concern.
+
+    Parameters
+    ----------
+    x : array_like, compatible object
+        If `x` is a list or tuple, it is converted to an ndarray, otherwise
+        it is left unchanged and treated as a scalar. In either case, `x`
+        or its elements must support addition and multiplication with
+        themselves and with the elements of `c`.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree n are contained in c[n]. If `c` is multidimensional the
+        remaining indices enumerate multiple polynomials. In the two
+        dimensional case the coefficients may be thought of as stored in
+        the columns of `c`.
+    tensor : boolean, optional
+        If True, the shape of the coefficient array is extended with ones
+        on the right, one for each dimension of `x`. Scalars have dimension 0
+        for this action. The result is that every column of coefficients in
+        `c` is evaluated for every element of `x`. If False, `x` is broadcast
+        over the columns of `c` for the evaluation.  This keyword is useful
+        when `c` is multidimensional. The default value is True.
+
+    Returns
+    -------
+    values : ndarray, algebra_like
+        The shape of the return value is described above.
+
+    See Also
+    --------
+    chebval2d, chebgrid2d, chebval3d, chebgrid3d
+
+    Notes
+    -----
+    The evaluation uses Clenshaw recursion, aka synthetic division.
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if isinstance(x, (tuple, list)):
+        x = np.asarray(x)
+    if isinstance(x, np.ndarray) and tensor:
+        c = c.reshape(c.shape + (1,) * x.ndim)
+
+    if len(c) == 1:
+        c0 = c[0]
+        c1 = 0
+    elif len(c) == 2:
+        c0 = c[0]
+        c1 = c[1]
+    else:
+        x2 = 2 * x
+        c0 = c[-2]
+        c1 = c[-1]
+        for i in range(3, len(c) + 1):
+            tmp = c0
+            c0 = c[-i] - c1
+            c1 = tmp + c1 * x2
+    return c0 + c1 * x
+
+
+def chebval2d(x, y, c):
+    """
+    Evaluate a 2-D Chebyshev series at points (x, y).
+
+    This function returns the values:
+
+    .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * T_i(x) * T_j(y)
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars and they
+    must have the same shape after conversion. In either case, either `x`
+    and `y` or their elements must support multiplication and addition both
+    with themselves and with the elements of `c`.
+
+    If `c` is a 1-D array a one is implicitly appended to its shape to make
+    it 2-D. The shape of the result will be c.shape[2:] + x.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points ``(x, y)``,
+        where `x` and `y` must have the same shape. If `x` or `y` is a list
+        or tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and if it isn't an ndarray it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term
+        of multi-degree i,j is contained in ``c[i,j]``. If `c` has
+        dimension greater than 2 the remaining indices enumerate multiple
+        sets of coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional Chebyshev series at points formed
+        from pairs of corresponding values from `x` and `y`.
+
+    See Also
+    --------
+    chebval, chebgrid2d, chebval3d, chebgrid3d
+    """
+    return pu._valnd(chebval, c, x, y)
+
+
+def chebgrid2d(x, y, c):
+    """
+    Evaluate a 2-D Chebyshev series on the Cartesian product of x and y.
+
+    This function returns the values:
+
+    .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * T_i(a) * T_j(b),
+
+    where the points `(a, b)` consist of all pairs formed by taking
+    `a` from `x` and `b` from `y`. The resulting points form a grid with
+    `x` in the first dimension and `y` in the second.
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars. In either
+    case, either `x` and `y` or their elements must support multiplication
+    and addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than two dimensions, ones are implicitly appended to
+    its shape to make it 2-D. The shape of the result will be c.shape[2:] +
+    x.shape + y.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points in the
+        Cartesian product of `x` and `y`.  If `x` or `y` is a list or
+        tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and, if it isn't an ndarray, it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term of
+        multi-degree i,j is contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional Chebyshev series at points in the
+        Cartesian product of `x` and `y`.
+
+    See Also
+    --------
+    chebval, chebval2d, chebval3d, chebgrid3d
+    """
+    return pu._gridnd(chebval, c, x, y)
+
+
+def chebval3d(x, y, z, c):
+    """
+    Evaluate a 3-D Chebyshev series at points (x, y, z).
+
+    This function returns the values:
+
+    .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * T_i(x) * T_j(y) * T_k(z)
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if
+    they are tuples or a lists, otherwise they are treated as a scalars and
+    they must have the same shape after conversion. In either case, either
+    `x`, `y`, and `z` or their elements must support multiplication and
+    addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than 3 dimensions, ones are implicitly appended to its
+    shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible object
+        The three dimensional series is evaluated at the points
+        ``(x, y, z)``, where `x`, `y`, and `z` must have the same shape.  If
+        any of `x`, `y`, or `z` is a list or tuple, it is first converted
+        to an ndarray, otherwise it is left unchanged and if it isn't an
+        ndarray it is  treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term of
+        multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
+        greater than 3 the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the multidimensional polynomial on points formed with
+        triples of corresponding values from `x`, `y`, and `z`.
+
+    See Also
+    --------
+    chebval, chebval2d, chebgrid2d, chebgrid3d
+    """
+    return pu._valnd(chebval, c, x, y, z)
+
+
+def chebgrid3d(x, y, z, c):
+    """
+    Evaluate a 3-D Chebyshev series on the Cartesian product of x, y, and z.
+
+    This function returns the values:
+
+    .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * T_i(a) * T_j(b) * T_k(c)
+
+    where the points ``(a, b, c)`` consist of all triples formed by taking
+    `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
+    a grid with `x` in the first dimension, `y` in the second, and `z` in
+    the third.
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if they
+    are tuples or a lists, otherwise they are treated as a scalars. In
+    either case, either `x`, `y`, and `z` or their elements must support
+    multiplication and addition both with themselves and with the elements
+    of `c`.
+
+    If `c` has fewer than three dimensions, ones are implicitly appended to
+    its shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape + y.shape + z.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible objects
+        The three dimensional series is evaluated at the points in the
+        Cartesian product of `x`, `y`, and `z`.  If `x`, `y`, or `z` is a
+        list or tuple, it is first converted to an ndarray, otherwise it is
+        left unchanged and, if it isn't an ndarray, it is treated as a
+        scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree i,j are contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points in the Cartesian
+        product of `x` and `y`.
+
+    See Also
+    --------
+    chebval, chebval2d, chebgrid2d, chebval3d
+    """
+    return pu._gridnd(chebval, c, x, y, z)
+
+
+def chebvander(x, deg):
+    """Pseudo-Vandermonde matrix of given degree.
+
+    Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
+    `x`. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., i] = T_i(x),
+
+    where ``0 <= i <= deg``. The leading indices of `V` index the elements of
+    `x` and the last index is the degree of the Chebyshev polynomial.
+
+    If `c` is a 1-D array of coefficients of length ``n + 1`` and `V` is the
+    matrix ``V = chebvander(x, n)``, then ``np.dot(V, c)`` and
+    ``chebval(x, c)`` are the same up to roundoff.  This equivalence is
+    useful both for least squares fitting and for the evaluation of a large
+    number of Chebyshev series of the same degree and sample points.
+
+    Parameters
+    ----------
+    x : array_like
+        Array of points. The dtype is converted to float64 or complex128
+        depending on whether any of the elements are complex. If `x` is
+        scalar it is converted to a 1-D array.
+    deg : int
+        Degree of the resulting matrix.
+
+    Returns
+    -------
+    vander : ndarray
+        The pseudo Vandermonde matrix. The shape of the returned matrix is
+        ``x.shape + (deg + 1,)``, where The last index is the degree of the
+        corresponding Chebyshev polynomial.  The dtype will be the same as
+        the converted `x`.
+
+    """
+    ideg = pu._as_int(deg, "deg")
+    if ideg < 0:
+        raise ValueError("deg must be non-negative")
+
+    x = np.array(x, copy=None, ndmin=1) + 0.0
+    dims = (ideg + 1,) + x.shape
+    dtyp = x.dtype
+    v = np.empty(dims, dtype=dtyp)
+    # Use forward recursion to generate the entries.
+    v[0] = x * 0 + 1
+    if ideg > 0:
+        x2 = 2 * x
+        v[1] = x
+        for i in range(2, ideg + 1):
+            v[i] = v[i - 1] * x2 - v[i - 2]
+    return np.moveaxis(v, 0, -1)
+
+
+def chebvander2d(x, y, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points ``(x, y)``. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (deg[1] + 1)*i + j] = T_i(x) * T_j(y),
+
+    where ``0 <= i <= deg[0]`` and ``0 <= j <= deg[1]``. The leading indices of
+    `V` index the points ``(x, y)`` and the last index encodes the degrees of
+    the Chebyshev polynomials.
+
+    If ``V = chebvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
+    correspond to the elements of a 2-D coefficient array `c` of shape
+    (xdeg + 1, ydeg + 1) in the order
+
+    .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
+
+    and ``np.dot(V, c.flat)`` and ``chebval2d(x, y, c)`` will be the same
+    up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 2-D Chebyshev
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes
+        will be converted to either float64 or complex128 depending on
+        whether any of the elements are complex. Scalars are converted to
+        1-D arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg].
+
+    Returns
+    -------
+    vander2d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)`.  The dtype will be the same
+        as the converted `x` and `y`.
+
+    See Also
+    --------
+    chebvander, chebvander3d, chebval2d, chebval3d
+    """
+    return pu._vander_nd_flat((chebvander, chebvander), (x, y), deg)
+
+
+def chebvander3d(x, y, z, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points ``(x, y, z)``. If `l`, `m`, `n` are the given degrees in `x`, `y`, `z`,
+    then The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = T_i(x)*T_j(y)*T_k(z),
+
+    where ``0 <= i <= l``, ``0 <= j <= m``, and ``0 <= j <= n``.  The leading
+    indices of `V` index the points ``(x, y, z)`` and the last index encodes
+    the degrees of the Chebyshev polynomials.
+
+    If ``V = chebvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
+    of `V` correspond to the elements of a 3-D coefficient array `c` of
+    shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
+
+    .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
+
+    and ``np.dot(V, c.flat)`` and ``chebval3d(x, y, z, c)`` will be the
+    same up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 3-D Chebyshev
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y, z : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes will
+        be converted to either float64 or complex128 depending on whether
+        any of the elements are complex. Scalars are converted to 1-D
+        arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg, z_deg].
+
+    Returns
+    -------
+    vander3d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`.  The dtype will
+        be the same as the converted `x`, `y`, and `z`.
+
+    See Also
+    --------
+    chebvander, chebvander3d, chebval2d, chebval3d
+    """
+    return pu._vander_nd_flat((chebvander, chebvander, chebvander), (x, y, z), deg)
+
+
+def chebfit(x, y, deg, rcond=None, full=False, w=None):
+    """
+    Least squares fit of Chebyshev series to data.
+
+    Return the coefficients of a Chebyshev series of degree `deg` that is the
+    least squares fit to the data values `y` given at points `x`. If `y` is
+    1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
+    fits are done, one for each column of `y`, and the resulting
+    coefficients are stored in the corresponding columns of a 2-D return.
+    The fitted polynomial(s) are in the form
+
+    .. math::  p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x),
+
+    where `n` is `deg`.
+
+    Parameters
+    ----------
+    x : array_like, shape (M,)
+        x-coordinates of the M sample points ``(x[i], y[i])``.
+    y : array_like, shape (M,) or (M, K)
+        y-coordinates of the sample points. Several data sets of sample
+        points sharing the same x-coordinates can be fitted at once by
+        passing in a 2D-array that contains one dataset per column.
+    deg : int or 1-D array_like
+        Degree(s) of the fitting polynomials. If `deg` is a single integer,
+        all terms up to and including the `deg`'th term are included in the
+        fit. For NumPy versions >= 1.11.0 a list of integers specifying the
+        degrees of the terms to include may be used instead.
+    rcond : float, optional
+        Relative condition number of the fit. Singular values smaller than
+        this relative to the largest singular value will be ignored. The
+        default value is ``len(x)*eps``, where eps is the relative precision of
+        the float type, about 2e-16 in most cases.
+    full : bool, optional
+        Switch determining nature of return value. When it is False (the
+        default) just the coefficients are returned, when True diagnostic
+        information from the singular value decomposition is also returned.
+    w : array_like, shape (`M`,), optional
+        Weights. If not None, the weight ``w[i]`` applies to the unsquared
+        residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
+        chosen so that the errors of the products ``w[i]*y[i]`` all have the
+        same variance.  When using inverse-variance weighting, use
+        ``w[i] = 1/sigma(y[i])``.  The default value is None.
+
+    Returns
+    -------
+    coef : ndarray, shape (M,) or (M, K)
+        Chebyshev coefficients ordered from low to high. If `y` was 2-D,
+        the coefficients for the data in column k  of `y` are in column
+        `k`.
+
+    [residuals, rank, singular_values, rcond] : list
+        These values are only returned if ``full == True``
+
+        - residuals -- sum of squared residuals of the least squares fit
+        - rank -- the numerical rank of the scaled Vandermonde matrix
+        - singular_values -- singular values of the scaled Vandermonde matrix
+        - rcond -- value of `rcond`.
+
+        For more details, see `numpy.linalg.lstsq`.
+
+    Warns
+    -----
+    RankWarning
+        The rank of the coefficient matrix in the least-squares fit is
+        deficient. The warning is only raised if ``full == False``.  The
+        warnings can be turned off by
+
+        >>> import warnings
+        >>> warnings.simplefilter('ignore', np.exceptions.RankWarning)
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyfit
+    numpy.polynomial.legendre.legfit
+    numpy.polynomial.laguerre.lagfit
+    numpy.polynomial.hermite.hermfit
+    numpy.polynomial.hermite_e.hermefit
+    chebval : Evaluates a Chebyshev series.
+    chebvander : Vandermonde matrix of Chebyshev series.
+    chebweight : Chebyshev weight function.
+    numpy.linalg.lstsq : Computes a least-squares fit from the matrix.
+    scipy.interpolate.UnivariateSpline : Computes spline fits.
+
+    Notes
+    -----
+    The solution is the coefficients of the Chebyshev series `p` that
+    minimizes the sum of the weighted squared errors
+
+    .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
+
+    where :math:`w_j` are the weights. This problem is solved by setting up
+    as the (typically) overdetermined matrix equation
+
+    .. math:: V(x) * c = w * y,
+
+    where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the
+    coefficients to be solved for, `w` are the weights, and `y` are the
+    observed values.  This equation is then solved using the singular value
+    decomposition of `V`.
+
+    If some of the singular values of `V` are so small that they are
+    neglected, then a `~exceptions.RankWarning` will be issued. This means that
+    the coefficient values may be poorly determined. Using a lower order fit
+    will usually get rid of the warning.  The `rcond` parameter can also be
+    set to a value smaller than its default, but the resulting fit may be
+    spurious and have large contributions from roundoff error.
+
+    Fits using Chebyshev series are usually better conditioned than fits
+    using power series, but much can depend on the distribution of the
+    sample points and the smoothness of the data. If the quality of the fit
+    is inadequate splines may be a good alternative.
+
+    References
+    ----------
+    .. [1] Wikipedia, "Curve fitting",
+           https://en.wikipedia.org/wiki/Curve_fitting
+
+    Examples
+    --------
+
+    """
+    return pu._fit(chebvander, x, y, deg, rcond, full, w)
+
+
+def chebcompanion(c):
+    """Return the scaled companion matrix of c.
+
+    The basis polynomials are scaled so that the companion matrix is
+    symmetric when `c` is a Chebyshev basis polynomial. This provides
+    better eigenvalue estimates than the unscaled case and for basis
+    polynomials the eigenvalues are guaranteed to be real if
+    `numpy.linalg.eigvalsh` is used to obtain them.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Chebyshev series coefficients ordered from low to high
+        degree.
+
+    Returns
+    -------
+    mat : ndarray
+        Scaled companion matrix of dimensions (deg, deg).
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) < 2:
+        raise ValueError('Series must have maximum degree of at least 1.')
+    if len(c) == 2:
+        return np.array([[-c[0] / c[1]]])
+
+    n = len(c) - 1
+    mat = np.zeros((n, n), dtype=c.dtype)
+    scl = np.array([1.] + [np.sqrt(.5)] * (n - 1))
+    top = mat.reshape(-1)[1::n + 1]
+    bot = mat.reshape(-1)[n::n + 1]
+    top[0] = np.sqrt(.5)
+    top[1:] = 1 / 2
+    bot[...] = top
+    mat[:, -1] -= (c[:-1] / c[-1]) * (scl / scl[-1]) * .5
+    return mat
+
+
+def chebroots(c):
+    """
+    Compute the roots of a Chebyshev series.
+
+    Return the roots (a.k.a. "zeros") of the polynomial
+
+    .. math:: p(x) = \\sum_i c[i] * T_i(x).
+
+    Parameters
+    ----------
+    c : 1-D array_like
+        1-D array of coefficients.
+
+    Returns
+    -------
+    out : ndarray
+        Array of the roots of the series. If all the roots are real,
+        then `out` is also real, otherwise it is complex.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyroots
+    numpy.polynomial.legendre.legroots
+    numpy.polynomial.laguerre.lagroots
+    numpy.polynomial.hermite.hermroots
+    numpy.polynomial.hermite_e.hermeroots
+
+    Notes
+    -----
+    The root estimates are obtained as the eigenvalues of the companion
+    matrix, Roots far from the origin of the complex plane may have large
+    errors due to the numerical instability of the series for such
+    values. Roots with multiplicity greater than 1 will also show larger
+    errors as the value of the series near such points is relatively
+    insensitive to errors in the roots. Isolated roots near the origin can
+    be improved by a few iterations of Newton's method.
+
+    The Chebyshev series basis polynomials aren't powers of `x` so the
+    results of this function may seem unintuitive.
+
+    Examples
+    --------
+    >>> import numpy.polynomial.chebyshev as cheb
+    >>> cheb.chebroots((-1, 1,-1, 1)) # T3 - T2 + T1 - T0 has real roots
+    array([ -5.00000000e-01,   2.60860684e-17,   1.00000000e+00]) # may vary
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) < 2:
+        return np.array([], dtype=c.dtype)
+    if len(c) == 2:
+        return np.array([-c[0] / c[1]])
+
+    # rotated companion matrix reduces error
+    m = chebcompanion(c)[::-1, ::-1]
+    r = np.linalg.eigvals(m)
+    r.sort()
+    return r
+
+
+def chebinterpolate(func, deg, args=()):
+    """Interpolate a function at the Chebyshev points of the first kind.
+
+    Returns the Chebyshev series that interpolates `func` at the Chebyshev
+    points of the first kind in the interval [-1, 1]. The interpolating
+    series tends to a minmax approximation to `func` with increasing `deg`
+    if the function is continuous in the interval.
+
+    Parameters
+    ----------
+    func : function
+        The function to be approximated. It must be a function of a single
+        variable of the form ``f(x, a, b, c...)``, where ``a, b, c...`` are
+        extra arguments passed in the `args` parameter.
+    deg : int
+        Degree of the interpolating polynomial
+    args : tuple, optional
+        Extra arguments to be used in the function call. Default is no extra
+        arguments.
+
+    Returns
+    -------
+    coef : ndarray, shape (deg + 1,)
+        Chebyshev coefficients of the interpolating series ordered from low to
+        high.
+
+    Examples
+    --------
+    >>> import numpy.polynomial.chebyshev as C
+    >>> C.chebinterpolate(lambda x: np.tanh(x) + 0.5, 8)
+    array([  5.00000000e-01,   8.11675684e-01,  -9.86864911e-17,
+            -5.42457905e-02,  -2.71387850e-16,   4.51658839e-03,
+             2.46716228e-17,  -3.79694221e-04,  -3.26899002e-16])
+
+    Notes
+    -----
+    The Chebyshev polynomials used in the interpolation are orthogonal when
+    sampled at the Chebyshev points of the first kind. If it is desired to
+    constrain some of the coefficients they can simply be set to the desired
+    value after the interpolation, no new interpolation or fit is needed. This
+    is especially useful if it is known apriori that some of coefficients are
+    zero. For instance, if the function is even then the coefficients of the
+    terms of odd degree in the result can be set to zero.
+
+    """
+    deg = np.asarray(deg)
+
+    # check arguments.
+    if deg.ndim > 0 or deg.dtype.kind not in 'iu' or deg.size == 0:
+        raise TypeError("deg must be an int")
+    if deg < 0:
+        raise ValueError("expected deg >= 0")
+
+    order = deg + 1
+    xcheb = chebpts1(order)
+    yfunc = func(xcheb, *args)
+    m = chebvander(xcheb, deg)
+    c = np.dot(m.T, yfunc)
+    c[0] /= order
+    c[1:] /= 0.5 * order
+
+    return c
+
+
+def chebgauss(deg):
+    """
+    Gauss-Chebyshev quadrature.
+
+    Computes the sample points and weights for Gauss-Chebyshev quadrature.
+    These sample points and weights will correctly integrate polynomials of
+    degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with
+    the weight function :math:`f(x) = 1/\\sqrt{1 - x^2}`.
+
+    Parameters
+    ----------
+    deg : int
+        Number of sample points and weights. It must be >= 1.
+
+    Returns
+    -------
+    x : ndarray
+        1-D ndarray containing the sample points.
+    y : ndarray
+        1-D ndarray containing the weights.
+
+    Notes
+    -----
+    The results have only been tested up to degree 100, higher degrees may
+    be problematic. For Gauss-Chebyshev there are closed form solutions for
+    the sample points and weights. If n = `deg`, then
+
+    .. math:: x_i = \\cos(\\pi (2 i - 1) / (2 n))
+
+    .. math:: w_i = \\pi / n
+
+    """
+    ideg = pu._as_int(deg, "deg")
+    if ideg <= 0:
+        raise ValueError("deg must be a positive integer")
+
+    x = np.cos(np.pi * np.arange(1, 2 * ideg, 2) / (2.0 * ideg))
+    w = np.ones(ideg) * (np.pi / ideg)
+
+    return x, w
+
+
+def chebweight(x):
+    """
+    The weight function of the Chebyshev polynomials.
+
+    The weight function is :math:`1/\\sqrt{1 - x^2}` and the interval of
+    integration is :math:`[-1, 1]`. The Chebyshev polynomials are
+    orthogonal, but not normalized, with respect to this weight function.
+
+    Parameters
+    ----------
+    x : array_like
+       Values at which the weight function will be computed.
+
+    Returns
+    -------
+    w : ndarray
+       The weight function at `x`.
+    """
+    w = 1. / (np.sqrt(1. + x) * np.sqrt(1. - x))
+    return w
+
+
+def chebpts1(npts):
+    """
+    Chebyshev points of the first kind.
+
+    The Chebyshev points of the first kind are the points ``cos(x)``,
+    where ``x = [pi*(k + .5)/npts for k in range(npts)]``.
+
+    Parameters
+    ----------
+    npts : int
+        Number of sample points desired.
+
+    Returns
+    -------
+    pts : ndarray
+        The Chebyshev points of the first kind.
+
+    See Also
+    --------
+    chebpts2
+    """
+    _npts = int(npts)
+    if _npts != npts:
+        raise ValueError("npts must be integer")
+    if _npts < 1:
+        raise ValueError("npts must be >= 1")
+
+    x = 0.5 * np.pi / _npts * np.arange(-_npts + 1, _npts + 1, 2)
+    return np.sin(x)
+
+
+def chebpts2(npts):
+    """
+    Chebyshev points of the second kind.
+
+    The Chebyshev points of the second kind are the points ``cos(x)``,
+    where ``x = [pi*k/(npts - 1) for k in range(npts)]`` sorted in ascending
+    order.
+
+    Parameters
+    ----------
+    npts : int
+        Number of sample points desired.
+
+    Returns
+    -------
+    pts : ndarray
+        The Chebyshev points of the second kind.
+    """
+    _npts = int(npts)
+    if _npts != npts:
+        raise ValueError("npts must be integer")
+    if _npts < 2:
+        raise ValueError("npts must be >= 2")
+
+    x = np.linspace(-np.pi, 0, _npts)
+    return np.cos(x)
+
+
+#
+# Chebyshev series class
+#
+
+class Chebyshev(ABCPolyBase):
+    """A Chebyshev series class.
+
+    The Chebyshev class provides the standard Python numerical methods
+    '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the
+    attributes and methods listed below.
+
+    Parameters
+    ----------
+    coef : array_like
+        Chebyshev coefficients in order of increasing degree, i.e.,
+        ``(1, 2, 3)`` gives ``1*T_0(x) + 2*T_1(x) + 3*T_2(x)``.
+    domain : (2,) array_like, optional
+        Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
+        to the interval ``[window[0], window[1]]`` by shifting and scaling.
+        The default value is [-1., 1.].
+    window : (2,) array_like, optional
+        Window, see `domain` for its use. The default value is [-1., 1.].
+    symbol : str, optional
+        Symbol used to represent the independent variable in string
+        representations of the polynomial expression, e.g. for printing.
+        The symbol must be a valid Python identifier. Default value is 'x'.
+
+        .. versionadded:: 1.24
+
+    """
+    # Virtual Functions
+    _add = staticmethod(chebadd)
+    _sub = staticmethod(chebsub)
+    _mul = staticmethod(chebmul)
+    _div = staticmethod(chebdiv)
+    _pow = staticmethod(chebpow)
+    _val = staticmethod(chebval)
+    _int = staticmethod(chebint)
+    _der = staticmethod(chebder)
+    _fit = staticmethod(chebfit)
+    _line = staticmethod(chebline)
+    _roots = staticmethod(chebroots)
+    _fromroots = staticmethod(chebfromroots)
+
+    @classmethod
+    def interpolate(cls, func, deg, domain=None, args=()):
+        """Interpolate a function at the Chebyshev points of the first kind.
+
+        Returns the series that interpolates `func` at the Chebyshev points of
+        the first kind scaled and shifted to the `domain`. The resulting series
+        tends to a minmax approximation of `func` when the function is
+        continuous in the domain.
+
+        Parameters
+        ----------
+        func : function
+            The function to be interpolated. It must be a function of a single
+            variable of the form ``f(x, a, b, c...)``, where ``a, b, c...`` are
+            extra arguments passed in the `args` parameter.
+        deg : int
+            Degree of the interpolating polynomial.
+        domain : {None, [beg, end]}, optional
+            Domain over which `func` is interpolated. The default is None, in
+            which case the domain is [-1, 1].
+        args : tuple, optional
+            Extra arguments to be used in the function call. Default is no
+            extra arguments.
+
+        Returns
+        -------
+        polynomial : Chebyshev instance
+            Interpolating Chebyshev instance.
+
+        Notes
+        -----
+        See `numpy.polynomial.chebinterpolate` for more details.
+
+        """
+        if domain is None:
+            domain = cls.domain
+        xfunc = lambda x: func(pu.mapdomain(x, cls.window, domain), *args)
+        coef = chebinterpolate(xfunc, deg)
+        return cls(coef, domain=domain)
+
+    # Virtual properties
+    domain = np.array(chebdomain)
+    window = np.array(chebdomain)
+    basis_name = 'T'

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy/polynomial/chebyshev.pyi

@@ -0,0 +1,180 @@
+from _typeshed import ConvertibleToInt
+from collections.abc import Callable, Iterable
+from typing import (
+    Any,
+    ClassVar,
+    Concatenate,
+    Final,
+    Literal as L,
+    Self,
+    TypeVar,
+    overload,
+)
+
+import numpy as np
+import numpy.typing as npt
+from numpy._typing import _IntLike_co
+
+from ._polybase import ABCPolyBase
+from ._polytypes import (
+    _Array1,
+    _Array2,
+    _CoefSeries,
+    _FuncBinOp,
+    _FuncCompanion,
+    _FuncDer,
+    _FuncFit,
+    _FuncFromRoots,
+    _FuncGauss,
+    _FuncInteg,
+    _FuncLine,
+    _FuncPoly2Ortho,
+    _FuncPow,
+    _FuncRoots,
+    _FuncUnOp,
+    _FuncVal,
+    _FuncVal2D,
+    _FuncVal3D,
+    _FuncVander,
+    _FuncVander2D,
+    _FuncVander3D,
+    _FuncWeight,
+    _Series,
+    _SeriesLikeCoef_co,
+)
+from .polyutils import trimcoef as chebtrim
+
+__all__ = [
+    "chebzero",
+    "chebone",
+    "chebx",
+    "chebdomain",
+    "chebline",
+    "chebadd",
+    "chebsub",
+    "chebmulx",
+    "chebmul",
+    "chebdiv",
+    "chebpow",
+    "chebval",
+    "chebder",
+    "chebint",
+    "cheb2poly",
+    "poly2cheb",
+    "chebfromroots",
+    "chebvander",
+    "chebfit",
+    "chebtrim",
+    "chebroots",
+    "chebpts1",
+    "chebpts2",
+    "Chebyshev",
+    "chebval2d",
+    "chebval3d",
+    "chebgrid2d",
+    "chebgrid3d",
+    "chebvander2d",
+    "chebvander3d",
+    "chebcompanion",
+    "chebgauss",
+    "chebweight",
+    "chebinterpolate",
+]
+
+_NumberOrObjectT = TypeVar("_NumberOrObjectT", bound=np.number | np.object_)
+_CoefScalarT = TypeVar("_CoefScalarT", bound=np.number | np.bool | np.object_)
+
+def _cseries_to_zseries(c: npt.NDArray[_NumberOrObjectT]) -> _Series[_NumberOrObjectT]: ...
+def _zseries_to_cseries(zs: npt.NDArray[_NumberOrObjectT]) -> _Series[_NumberOrObjectT]: ...
+def _zseries_mul(z1: npt.NDArray[_NumberOrObjectT], z2: npt.NDArray[_NumberOrObjectT]) -> _Series[_NumberOrObjectT]: ...
+def _zseries_div(z1: npt.NDArray[_NumberOrObjectT], z2: npt.NDArray[_NumberOrObjectT]) -> _Series[_NumberOrObjectT]: ...
+def _zseries_der(zs: npt.NDArray[_NumberOrObjectT]) -> _Series[_NumberOrObjectT]: ...
+def _zseries_int(zs: npt.NDArray[_NumberOrObjectT]) -> _Series[_NumberOrObjectT]: ...
+
+poly2cheb: Final[_FuncPoly2Ortho] = ...
+cheb2poly: Final[_FuncUnOp] = ...
+
+chebdomain: Final[_Array2[np.float64]] = ...
+chebzero: Final[_Array1[np.int_]] = ...
+chebone: Final[_Array1[np.int_]] = ...
+chebx: Final[_Array2[np.int_]] = ...
+
+chebline: Final[_FuncLine] = ...
+chebfromroots: Final[_FuncFromRoots] = ...
+chebadd: Final[_FuncBinOp] = ...
+chebsub: Final[_FuncBinOp] = ...
+chebmulx: Final[_FuncUnOp] = ...
+chebmul: Final[_FuncBinOp] = ...
+chebdiv: Final[_FuncBinOp] = ...
+chebpow: Final[_FuncPow] = ...
+chebder: Final[_FuncDer] = ...
+chebint: Final[_FuncInteg] = ...
+chebval: Final[_FuncVal] = ...
+chebval2d: Final[_FuncVal2D] = ...
+chebval3d: Final[_FuncVal3D] = ...
+chebgrid2d: Final[_FuncVal2D] = ...
+chebgrid3d: Final[_FuncVal3D] = ...
+chebvander: Final[_FuncVander] = ...
+chebvander2d: Final[_FuncVander2D] = ...
+chebvander3d: Final[_FuncVander3D] = ...
+chebfit: Final[_FuncFit] = ...
+chebcompanion: Final[_FuncCompanion] = ...
+chebroots: Final[_FuncRoots] = ...
+chebgauss: Final[_FuncGauss] = ...
+chebweight: Final[_FuncWeight] = ...
+def chebpts1(npts: ConvertibleToInt) -> np.ndarray[tuple[int], np.dtype[np.float64]]: ...
+def chebpts2(npts: ConvertibleToInt) -> np.ndarray[tuple[int], np.dtype[np.float64]]: ...
+
+# keep in sync with `Chebyshev.interpolate` (minus `domain` parameter)
+@overload
+def chebinterpolate(
+    func: np.ufunc,
+    deg: _IntLike_co,
+    args: tuple[()] = (),
+) -> npt.NDArray[np.float64 | np.complex128 | np.object_]: ...
+@overload
+def chebinterpolate(
+    func: Callable[[npt.NDArray[np.float64]], _CoefScalarT],
+    deg: _IntLike_co,
+    args: tuple[()] = (),
+) -> npt.NDArray[_CoefScalarT]: ...
+@overload
+def chebinterpolate(
+    func: Callable[Concatenate[npt.NDArray[np.float64], ...], _CoefScalarT],
+    deg: _IntLike_co,
+    args: Iterable[Any],
+) -> npt.NDArray[_CoefScalarT]: ...
+
+class Chebyshev(ABCPolyBase[L["T"]]):
+    basis_name: ClassVar[L["T"]] = "T"  # pyright: ignore[reportIncompatibleMethodOverride]
+    domain: _Array2[np.float64 | Any] = ...  # pyright: ignore[reportIncompatibleMethodOverride]
+    window: _Array2[np.float64 | Any] = ...  # pyright: ignore[reportIncompatibleMethodOverride]
+
+    @overload
+    @classmethod
+    def interpolate(
+        cls,
+        func: Callable[[npt.NDArray[np.float64]], _CoefSeries],
+        deg: _IntLike_co,
+        domain: _SeriesLikeCoef_co | None = None,
+        args: tuple[()] = (),
+    ) -> Self: ...
+    @overload
+    @classmethod
+    def interpolate(
+        cls,
+        func: Callable[Concatenate[npt.NDArray[np.float64], ...], _CoefSeries],
+        deg: _IntLike_co,
+        domain: _SeriesLikeCoef_co | None = None,
+        *,
+        args: Iterable[Any],
+    ) -> Self: ...
+    @overload
+    @classmethod
+    def interpolate(
+        cls,
+        func: Callable[Concatenate[npt.NDArray[np.float64], ...], _CoefSeries],
+        deg: _IntLike_co,
+        domain: _SeriesLikeCoef_co | None,
+        args: Iterable[Any],
+    ) -> Self: ...

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy/polynomial/hermite.py

@@ -0,0 +1,1738 @@
+"""
+==============================================================
+Hermite Series, "Physicists" (:mod:`numpy.polynomial.hermite`)
+==============================================================
+
+This module provides a number of objects (mostly functions) useful for
+dealing with Hermite series, including a `Hermite` class that
+encapsulates the usual arithmetic operations.  (General information
+on how this module represents and works with such polynomials is in the
+docstring for its "parent" sub-package, `numpy.polynomial`).
+
+Classes
+-------
+.. autosummary::
+   :toctree: generated/
+
+   Hermite
+
+Constants
+---------
+.. autosummary::
+   :toctree: generated/
+
+   hermdomain
+   hermzero
+   hermone
+   hermx
+
+Arithmetic
+----------
+.. autosummary::
+   :toctree: generated/
+
+   hermadd
+   hermsub
+   hermmulx
+   hermmul
+   hermdiv
+   hermpow
+   hermval
+   hermval2d
+   hermval3d
+   hermgrid2d
+   hermgrid3d
+
+Calculus
+--------
+.. autosummary::
+   :toctree: generated/
+
+   hermder
+   hermint
+
+Misc Functions
+--------------
+.. autosummary::
+   :toctree: generated/
+
+   hermfromroots
+   hermroots
+   hermvander
+   hermvander2d
+   hermvander3d
+   hermgauss
+   hermweight
+   hermcompanion
+   hermfit
+   hermtrim
+   hermline
+   herm2poly
+   poly2herm
+
+See also
+--------
+`numpy.polynomial`
+
+"""
+import numpy as np
+
+from . import polyutils as pu
+from ._polybase import ABCPolyBase
+
+__all__ = [
+    'hermzero', 'hermone', 'hermx', 'hermdomain', 'hermline', 'hermadd',
+    'hermsub', 'hermmulx', 'hermmul', 'hermdiv', 'hermpow', 'hermval',
+    'hermder', 'hermint', 'herm2poly', 'poly2herm', 'hermfromroots',
+    'hermvander', 'hermfit', 'hermtrim', 'hermroots', 'Hermite',
+    'hermval2d', 'hermval3d', 'hermgrid2d', 'hermgrid3d', 'hermvander2d',
+    'hermvander3d', 'hermcompanion', 'hermgauss', 'hermweight']
+
+hermtrim = pu.trimcoef
+
+
+def poly2herm(pol):
+    """
+    poly2herm(pol)
+
+    Convert a polynomial to a Hermite series.
+
+    Convert an array representing the coefficients of a polynomial (relative
+    to the "standard" basis) ordered from lowest degree to highest, to an
+    array of the coefficients of the equivalent Hermite series, ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    pol : array_like
+        1-D array containing the polynomial coefficients
+
+    Returns
+    -------
+    c : ndarray
+        1-D array containing the coefficients of the equivalent Hermite
+        series.
+
+    See Also
+    --------
+    herm2poly
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import poly2herm
+    >>> poly2herm(np.arange(4))
+    array([1.   ,  2.75 ,  0.5  ,  0.375])
+
+    """
+    [pol] = pu.as_series([pol])
+    deg = len(pol) - 1
+    res = 0
+    for i in range(deg, -1, -1):
+        res = hermadd(hermmulx(res), pol[i])
+    return res
+
+
+def herm2poly(c):
+    """
+    Convert a Hermite series to a polynomial.
+
+    Convert an array representing the coefficients of a Hermite series,
+    ordered from lowest degree to highest, to an array of the coefficients
+    of the equivalent polynomial (relative to the "standard" basis) ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array containing the Hermite series coefficients, ordered
+        from lowest order term to highest.
+
+    Returns
+    -------
+    pol : ndarray
+        1-D array containing the coefficients of the equivalent polynomial
+        (relative to the "standard" basis) ordered from lowest order term
+        to highest.
+
+    See Also
+    --------
+    poly2herm
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import herm2poly
+    >>> herm2poly([ 1.   ,  2.75 ,  0.5  ,  0.375])
+    array([0., 1., 2., 3.])
+
+    """
+    from .polynomial import polyadd, polymulx, polysub
+
+    [c] = pu.as_series([c])
+    n = len(c)
+    if n == 1:
+        return c
+    if n == 2:
+        c[1] *= 2
+        return c
+    else:
+        c0 = c[-2]
+        c1 = c[-1]
+        # i is the current degree of c1
+        for i in range(n - 1, 1, -1):
+            tmp = c0
+            c0 = polysub(c[i - 2], c1 * (2 * (i - 1)))
+            c1 = polyadd(tmp, polymulx(c1) * 2)
+        return polyadd(c0, polymulx(c1) * 2)
+
+
+#
+# These are constant arrays are of integer type so as to be compatible
+# with the widest range of other types, such as Decimal.
+#
+
+# Hermite
+hermdomain = np.array([-1., 1.])
+
+# Hermite coefficients representing zero.
+hermzero = np.array([0])
+
+# Hermite coefficients representing one.
+hermone = np.array([1])
+
+# Hermite coefficients representing the identity x.
+hermx = np.array([0, 1 / 2])
+
+
+def hermline(off, scl):
+    """
+    Hermite series whose graph is a straight line.
+
+
+
+    Parameters
+    ----------
+    off, scl : scalars
+        The specified line is given by ``off + scl*x``.
+
+    Returns
+    -------
+    y : ndarray
+        This module's representation of the Hermite series for
+        ``off + scl*x``.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyline
+    numpy.polynomial.chebyshev.chebline
+    numpy.polynomial.legendre.legline
+    numpy.polynomial.laguerre.lagline
+    numpy.polynomial.hermite_e.hermeline
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermline, hermval
+    >>> hermval(0,hermline(3, 2))
+    3.0
+    >>> hermval(1,hermline(3, 2))
+    5.0
+
+    """
+    if scl != 0:
+        return np.array([off, scl / 2])
+    else:
+        return np.array([off])
+
+
+def hermfromroots(roots):
+    """
+    Generate a Hermite series with given roots.
+
+    The function returns the coefficients of the polynomial
+
+    .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
+
+    in Hermite form, where the :math:`r_n` are the roots specified in `roots`.
+    If a zero has multiplicity n, then it must appear in `roots` n times.
+    For instance, if 2 is a root of multiplicity three and 3 is a root of
+    multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
+    roots can appear in any order.
+
+    If the returned coefficients are `c`, then
+
+    .. math:: p(x) = c_0 + c_1 * H_1(x) + ... +  c_n * H_n(x)
+
+    The coefficient of the last term is not generally 1 for monic
+    polynomials in Hermite form.
+
+    Parameters
+    ----------
+    roots : array_like
+        Sequence containing the roots.
+
+    Returns
+    -------
+    out : ndarray
+        1-D array of coefficients.  If all roots are real then `out` is a
+        real array, if some of the roots are complex, then `out` is complex
+        even if all the coefficients in the result are real (see Examples
+        below).
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyfromroots
+    numpy.polynomial.legendre.legfromroots
+    numpy.polynomial.laguerre.lagfromroots
+    numpy.polynomial.chebyshev.chebfromroots
+    numpy.polynomial.hermite_e.hermefromroots
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermfromroots, hermval
+    >>> coef = hermfromroots((-1, 0, 1))
+    >>> hermval((-1, 0, 1), coef)
+    array([0.,  0.,  0.])
+    >>> coef = hermfromroots((-1j, 1j))
+    >>> hermval((-1j, 1j), coef)
+    array([0.+0.j, 0.+0.j])
+
+    """
+    return pu._fromroots(hermline, hermmul, roots)
+
+
+def hermadd(c1, c2):
+    """
+    Add one Hermite series to another.
+
+    Returns the sum of two Hermite series `c1` + `c2`.  The arguments
+    are sequences of coefficients ordered from lowest order term to
+    highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the Hermite series of their sum.
+
+    See Also
+    --------
+    hermsub, hermmulx, hermmul, hermdiv, hermpow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the sum of two Hermite series
+    is a Hermite series (without having to "reproject" the result onto
+    the basis set) so addition, just like that of "standard" polynomials,
+    is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermadd
+    >>> hermadd([1, 2, 3], [1, 2, 3, 4])
+    array([2., 4., 6., 4.])
+
+    """
+    return pu._add(c1, c2)
+
+
+def hermsub(c1, c2):
+    """
+    Subtract one Hermite series from another.
+
+    Returns the difference of two Hermite series `c1` - `c2`.  The
+    sequences of coefficients are from lowest order term to highest, i.e.,
+    [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Hermite series coefficients representing their difference.
+
+    See Also
+    --------
+    hermadd, hermmulx, hermmul, hermdiv, hermpow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the difference of two Hermite
+    series is a Hermite series (without having to "reproject" the result
+    onto the basis set) so subtraction, just like that of "standard"
+    polynomials, is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermsub
+    >>> hermsub([1, 2, 3, 4], [1, 2, 3])
+    array([0.,  0.,  0.,  4.])
+
+    """
+    return pu._sub(c1, c2)
+
+
+def hermmulx(c):
+    """Multiply a Hermite series by x.
+
+    Multiply the Hermite series `c` by x, where x is the independent
+    variable.
+
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the result of the multiplication.
+
+    See Also
+    --------
+    hermadd, hermsub, hermmul, hermdiv, hermpow
+
+    Notes
+    -----
+    The multiplication uses the recursion relationship for Hermite
+    polynomials in the form
+
+    .. math::
+
+        xP_i(x) = (P_{i + 1}(x)/2 + i*P_{i - 1}(x))
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermmulx
+    >>> hermmulx([1, 2, 3])
+    array([2. , 6.5, 1. , 1.5])
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    # The zero series needs special treatment
+    if len(c) == 1 and c[0] == 0:
+        return c
+
+    prd = np.empty(len(c) + 1, dtype=c.dtype)
+    prd[0] = c[0] * 0
+    prd[1] = c[0] / 2
+    for i in range(1, len(c)):
+        prd[i + 1] = c[i] / 2
+        prd[i - 1] += c[i] * i
+    return prd
+
+
+def hermmul(c1, c2):
+    """
+    Multiply one Hermite series by another.
+
+    Returns the product of two Hermite series `c1` * `c2`.  The arguments
+    are sequences of coefficients, from lowest order "term" to highest,
+    e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Hermite series coefficients representing their product.
+
+    See Also
+    --------
+    hermadd, hermsub, hermmulx, hermdiv, hermpow
+
+    Notes
+    -----
+    In general, the (polynomial) product of two C-series results in terms
+    that are not in the Hermite polynomial basis set.  Thus, to express
+    the product as a Hermite series, it is necessary to "reproject" the
+    product onto said basis set, which may produce "unintuitive" (but
+    correct) results; see Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermmul
+    >>> hermmul([1, 2, 3], [0, 1, 2])
+    array([52.,  29.,  52.,   7.,   6.])
+
+    """
+    # s1, s2 are trimmed copies
+    [c1, c2] = pu.as_series([c1, c2])
+
+    if len(c1) > len(c2):
+        c = c2
+        xs = c1
+    else:
+        c = c1
+        xs = c2
+
+    if len(c) == 1:
+        c0 = c[0] * xs
+        c1 = 0
+    elif len(c) == 2:
+        c0 = c[0] * xs
+        c1 = c[1] * xs
+    else:
+        nd = len(c)
+        c0 = c[-2] * xs
+        c1 = c[-1] * xs
+        for i in range(3, len(c) + 1):
+            tmp = c0
+            nd = nd - 1
+            c0 = hermsub(c[-i] * xs, c1 * (2 * (nd - 1)))
+            c1 = hermadd(tmp, hermmulx(c1) * 2)
+    return hermadd(c0, hermmulx(c1) * 2)
+
+
+def hermdiv(c1, c2):
+    """
+    Divide one Hermite series by another.
+
+    Returns the quotient-with-remainder of two Hermite series
+    `c1` / `c2`.  The arguments are sequences of coefficients from lowest
+    order "term" to highest, e.g., [1,2,3] represents the series
+    ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    [quo, rem] : ndarrays
+        Of Hermite series coefficients representing the quotient and
+        remainder.
+
+    See Also
+    --------
+    hermadd, hermsub, hermmulx, hermmul, hermpow
+
+    Notes
+    -----
+    In general, the (polynomial) division of one Hermite series by another
+    results in quotient and remainder terms that are not in the Hermite
+    polynomial basis set.  Thus, to express these results as a Hermite
+    series, it is necessary to "reproject" the results onto the Hermite
+    basis set, which may produce "unintuitive" (but correct) results; see
+    Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermdiv
+    >>> hermdiv([ 52.,  29.,  52.,   7.,   6.], [0, 1, 2])
+    (array([1., 2., 3.]), array([0.]))
+    >>> hermdiv([ 54.,  31.,  52.,   7.,   6.], [0, 1, 2])
+    (array([1., 2., 3.]), array([2., 2.]))
+    >>> hermdiv([ 53.,  30.,  52.,   7.,   6.], [0, 1, 2])
+    (array([1., 2., 3.]), array([1., 1.]))
+
+    """
+    return pu._div(hermmul, c1, c2)
+
+
+def hermpow(c, pow, maxpower=16):
+    """Raise a Hermite series to a power.
+
+    Returns the Hermite series `c` raised to the power `pow`. The
+    argument `c` is a sequence of coefficients ordered from low to high.
+    i.e., [1,2,3] is the series  ``P_0 + 2*P_1 + 3*P_2.``
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Hermite series coefficients ordered from low to
+        high.
+    pow : integer
+        Power to which the series will be raised
+    maxpower : integer, optional
+        Maximum power allowed. This is mainly to limit growth of the series
+        to unmanageable size. Default is 16
+
+    Returns
+    -------
+    coef : ndarray
+        Hermite series of power.
+
+    See Also
+    --------
+    hermadd, hermsub, hermmulx, hermmul, hermdiv
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermpow
+    >>> hermpow([1, 2, 3], 2)
+    array([81.,  52.,  82.,  12.,   9.])
+
+    """
+    return pu._pow(hermmul, c, pow, maxpower)
+
+
+def hermder(c, m=1, scl=1, axis=0):
+    """
+    Differentiate a Hermite series.
+
+    Returns the Hermite series coefficients `c` differentiated `m` times
+    along `axis`.  At each iteration the result is multiplied by `scl` (the
+    scaling factor is for use in a linear change of variable). The argument
+    `c` is an array of coefficients from low to high degree along each
+    axis, e.g., [1,2,3] represents the series ``1*H_0 + 2*H_1 + 3*H_2``
+    while [[1,2],[1,2]] represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) +
+    2*H_0(x)*H_1(y) + 2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is
+    ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Hermite series coefficients. If `c` is multidimensional the
+        different axis correspond to different variables with the degree in
+        each axis given by the corresponding index.
+    m : int, optional
+        Number of derivatives taken, must be non-negative. (Default: 1)
+    scl : scalar, optional
+        Each differentiation is multiplied by `scl`.  The end result is
+        multiplication by ``scl**m``.  This is for use in a linear change of
+        variable. (Default: 1)
+    axis : int, optional
+        Axis over which the derivative is taken. (Default: 0).
+
+    Returns
+    -------
+    der : ndarray
+        Hermite series of the derivative.
+
+    See Also
+    --------
+    hermint
+
+    Notes
+    -----
+    In general, the result of differentiating a Hermite series does not
+    resemble the same operation on a power series. Thus the result of this
+    function may be "unintuitive," albeit correct; see Examples section
+    below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermder
+    >>> hermder([ 1. ,  0.5,  0.5,  0.5])
+    array([1., 2., 3.])
+    >>> hermder([-0.5,  1./2.,  1./8.,  1./12.,  1./16.], m=2)
+    array([1., 2., 3.])
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    cnt = pu._as_int(m, "the order of derivation")
+    iaxis = pu._as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of derivation must be non-negative")
+    iaxis = np.lib.array_utils.normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    n = len(c)
+    if cnt >= n:
+        c = c[:1] * 0
+    else:
+        for i in range(cnt):
+            n = n - 1
+            c *= scl
+            der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
+            for j in range(n, 0, -1):
+                der[j - 1] = (2 * j) * c[j]
+            c = der
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def hermint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
+    """
+    Integrate a Hermite series.
+
+    Returns the Hermite series coefficients `c` integrated `m` times from
+    `lbnd` along `axis`. At each iteration the resulting series is
+    **multiplied** by `scl` and an integration constant, `k`, is added.
+    The scaling factor is for use in a linear change of variable.  ("Buyer
+    beware": note that, depending on what one is doing, one may want `scl`
+    to be the reciprocal of what one might expect; for more information,
+    see the Notes section below.)  The argument `c` is an array of
+    coefficients from low to high degree along each axis, e.g., [1,2,3]
+    represents the series ``H_0 + 2*H_1 + 3*H_2`` while [[1,2],[1,2]]
+    represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) +
+    2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Hermite series coefficients. If c is multidimensional the
+        different axis correspond to different variables with the degree in
+        each axis given by the corresponding index.
+    m : int, optional
+        Order of integration, must be positive. (Default: 1)
+    k : {[], list, scalar}, optional
+        Integration constant(s).  The value of the first integral at
+        ``lbnd`` is the first value in the list, the value of the second
+        integral at ``lbnd`` is the second value, etc.  If ``k == []`` (the
+        default), all constants are set to zero.  If ``m == 1``, a single
+        scalar can be given instead of a list.
+    lbnd : scalar, optional
+        The lower bound of the integral. (Default: 0)
+    scl : scalar, optional
+        Following each integration the result is *multiplied* by `scl`
+        before the integration constant is added. (Default: 1)
+    axis : int, optional
+        Axis over which the integral is taken. (Default: 0).
+
+    Returns
+    -------
+    S : ndarray
+        Hermite series coefficients of the integral.
+
+    Raises
+    ------
+    ValueError
+        If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
+        ``np.ndim(scl) != 0``.
+
+    See Also
+    --------
+    hermder
+
+    Notes
+    -----
+    Note that the result of each integration is *multiplied* by `scl`.
+    Why is this important to note?  Say one is making a linear change of
+    variable :math:`u = ax + b` in an integral relative to `x`.  Then
+    :math:`dx = du/a`, so one will need to set `scl` equal to
+    :math:`1/a` - perhaps not what one would have first thought.
+
+    Also note that, in general, the result of integrating a C-series needs
+    to be "reprojected" onto the C-series basis set.  Thus, typically,
+    the result of this function is "unintuitive," albeit correct; see
+    Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermint
+    >>> hermint([1,2,3]) # integrate once, value 0 at 0.
+    array([1. , 0.5, 0.5, 0.5])
+    >>> hermint([1,2,3], m=2) # integrate twice, value & deriv 0 at 0
+    array([-0.5       ,  0.5       ,  0.125     ,  0.08333333,  0.0625    ]) # may vary
+    >>> hermint([1,2,3], k=1) # integrate once, value 1 at 0.
+    array([2. , 0.5, 0.5, 0.5])
+    >>> hermint([1,2,3], lbnd=-1) # integrate once, value 0 at -1
+    array([-2. ,  0.5,  0.5,  0.5])
+    >>> hermint([1,2,3], m=2, k=[1,2], lbnd=-1)
+    array([ 1.66666667, -0.5       ,  0.125     ,  0.08333333,  0.0625    ]) # may vary
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if not np.iterable(k):
+        k = [k]
+    cnt = pu._as_int(m, "the order of integration")
+    iaxis = pu._as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of integration must be non-negative")
+    if len(k) > cnt:
+        raise ValueError("Too many integration constants")
+    if np.ndim(lbnd) != 0:
+        raise ValueError("lbnd must be a scalar.")
+    if np.ndim(scl) != 0:
+        raise ValueError("scl must be a scalar.")
+    iaxis = np.lib.array_utils.normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    k = list(k) + [0] * (cnt - len(k))
+    for i in range(cnt):
+        n = len(c)
+        c *= scl
+        if n == 1 and np.all(c[0] == 0):
+            c[0] += k[i]
+        else:
+            tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype)
+            tmp[0] = c[0] * 0
+            tmp[1] = c[0] / 2
+            for j in range(1, n):
+                tmp[j + 1] = c[j] / (2 * (j + 1))
+            tmp[0] += k[i] - hermval(lbnd, tmp)
+            c = tmp
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def hermval(x, c, tensor=True):
+    """
+    Evaluate a Hermite series at points x.
+
+    If `c` is of length ``n + 1``, this function returns the value:
+
+    .. math:: p(x) = c_0 * H_0(x) + c_1 * H_1(x) + ... + c_n * H_n(x)
+
+    The parameter `x` is converted to an array only if it is a tuple or a
+    list, otherwise it is treated as a scalar. In either case, either `x`
+    or its elements must support multiplication and addition both with
+    themselves and with the elements of `c`.
+
+    If `c` is a 1-D array, then ``p(x)`` will have the same shape as `x`.  If
+    `c` is multidimensional, then the shape of the result depends on the
+    value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
+    x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
+    scalars have shape (,).
+
+    Trailing zeros in the coefficients will be used in the evaluation, so
+    they should be avoided if efficiency is a concern.
+
+    Parameters
+    ----------
+    x : array_like, compatible object
+        If `x` is a list or tuple, it is converted to an ndarray, otherwise
+        it is left unchanged and treated as a scalar. In either case, `x`
+        or its elements must support addition and multiplication with
+        themselves and with the elements of `c`.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree n are contained in c[n]. If `c` is multidimensional the
+        remaining indices enumerate multiple polynomials. In the two
+        dimensional case the coefficients may be thought of as stored in
+        the columns of `c`.
+    tensor : boolean, optional
+        If True, the shape of the coefficient array is extended with ones
+        on the right, one for each dimension of `x`. Scalars have dimension 0
+        for this action. The result is that every column of coefficients in
+        `c` is evaluated for every element of `x`. If False, `x` is broadcast
+        over the columns of `c` for the evaluation.  This keyword is useful
+        when `c` is multidimensional. The default value is True.
+
+    Returns
+    -------
+    values : ndarray, algebra_like
+        The shape of the return value is described above.
+
+    See Also
+    --------
+    hermval2d, hermgrid2d, hermval3d, hermgrid3d
+
+    Notes
+    -----
+    The evaluation uses Clenshaw recursion, aka synthetic division.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermval
+    >>> coef = [1,2,3]
+    >>> hermval(1, coef)
+    11.0
+    >>> hermval([[1,2],[3,4]], coef)
+    array([[ 11.,   51.],
+           [115.,  203.]])
+
+    """
+    c = np.array(c, ndmin=1, copy=None)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if isinstance(x, (tuple, list)):
+        x = np.asarray(x)
+    if isinstance(x, np.ndarray) and tensor:
+        c = c.reshape(c.shape + (1,) * x.ndim)
+
+    x2 = x * 2
+    if len(c) == 1:
+        c0 = c[0]
+        c1 = 0
+    elif len(c) == 2:
+        c0 = c[0]
+        c1 = c[1]
+    else:
+        nd = len(c)
+        c0 = c[-2]
+        c1 = c[-1]
+        for i in range(3, len(c) + 1):
+            tmp = c0
+            nd = nd - 1
+            c0 = c[-i] - c1 * (2 * (nd - 1))
+            c1 = tmp + c1 * x2
+    return c0 + c1 * x2
+
+
+def hermval2d(x, y, c):
+    """
+    Evaluate a 2-D Hermite series at points (x, y).
+
+    This function returns the values:
+
+    .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * H_i(x) * H_j(y)
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars and they
+    must have the same shape after conversion. In either case, either `x`
+    and `y` or their elements must support multiplication and addition both
+    with themselves and with the elements of `c`.
+
+    If `c` is a 1-D array a one is implicitly appended to its shape to make
+    it 2-D. The shape of the result will be c.shape[2:] + x.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points ``(x, y)``,
+        where `x` and `y` must have the same shape. If `x` or `y` is a list
+        or tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and if it isn't an ndarray it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term
+        of multi-degree i,j is contained in ``c[i,j]``. If `c` has
+        dimension greater than two the remaining indices enumerate multiple
+        sets of coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points formed with
+        pairs of corresponding values from `x` and `y`.
+
+    See Also
+    --------
+    hermval, hermgrid2d, hermval3d, hermgrid3d
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermval2d
+    >>> x = [1, 2]
+    >>> y = [4, 5]
+    >>> c = [[1, 2, 3], [4, 5, 6]]
+    >>> hermval2d(x, y, c)
+    array([1035., 2883.])
+
+    """
+    return pu._valnd(hermval, c, x, y)
+
+
+def hermgrid2d(x, y, c):
+    """
+    Evaluate a 2-D Hermite series on the Cartesian product of x and y.
+
+    This function returns the values:
+
+    .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * H_i(a) * H_j(b)
+
+    where the points ``(a, b)`` consist of all pairs formed by taking
+    `a` from `x` and `b` from `y`. The resulting points form a grid with
+    `x` in the first dimension and `y` in the second.
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars. In either
+    case, either `x` and `y` or their elements must support multiplication
+    and addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than two dimensions, ones are implicitly appended to
+    its shape to make it 2-D. The shape of the result will be c.shape[2:] +
+    x.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points in the
+        Cartesian product of `x` and `y`.  If `x` or `y` is a list or
+        tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and, if it isn't an ndarray, it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree i,j are contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points in the Cartesian
+        product of `x` and `y`.
+
+    See Also
+    --------
+    hermval, hermval2d, hermval3d, hermgrid3d
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermgrid2d
+    >>> x = [1, 2, 3]
+    >>> y = [4, 5]
+    >>> c = [[1, 2, 3], [4, 5, 6]]
+    >>> hermgrid2d(x, y, c)
+    array([[1035., 1599.],
+           [1867., 2883.],
+           [2699., 4167.]])
+
+    """
+    return pu._gridnd(hermval, c, x, y)
+
+
+def hermval3d(x, y, z, c):
+    """
+    Evaluate a 3-D Hermite series at points (x, y, z).
+
+    This function returns the values:
+
+    .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * H_i(x) * H_j(y) * H_k(z)
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if
+    they are tuples or a lists, otherwise they are treated as a scalars and
+    they must have the same shape after conversion. In either case, either
+    `x`, `y`, and `z` or their elements must support multiplication and
+    addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than 3 dimensions, ones are implicitly appended to its
+    shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible object
+        The three dimensional series is evaluated at the points
+        ``(x, y, z)``, where `x`, `y`, and `z` must have the same shape.  If
+        any of `x`, `y`, or `z` is a list or tuple, it is first converted
+        to an ndarray, otherwise it is left unchanged and if it isn't an
+        ndarray it is  treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term of
+        multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
+        greater than 3 the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the multidimensional polynomial on points formed with
+        triples of corresponding values from `x`, `y`, and `z`.
+
+    See Also
+    --------
+    hermval, hermval2d, hermgrid2d, hermgrid3d
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermval3d
+    >>> x = [1, 2]
+    >>> y = [4, 5]
+    >>> z = [6, 7]
+    >>> c = [[[1, 2, 3], [4, 5, 6]], [[7, 8, 9], [10, 11, 12]]]
+    >>> hermval3d(x, y, z, c)
+    array([ 40077., 120131.])
+
+    """
+    return pu._valnd(hermval, c, x, y, z)
+
+
+def hermgrid3d(x, y, z, c):
+    """
+    Evaluate a 3-D Hermite series on the Cartesian product of x, y, and z.
+
+    This function returns the values:
+
+    .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * H_i(a) * H_j(b) * H_k(c)
+
+    where the points ``(a, b, c)`` consist of all triples formed by taking
+    `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
+    a grid with `x` in the first dimension, `y` in the second, and `z` in
+    the third.
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if they
+    are tuples or a lists, otherwise they are treated as a scalars. In
+    either case, either `x`, `y`, and `z` or their elements must support
+    multiplication and addition both with themselves and with the elements
+    of `c`.
+
+    If `c` has fewer than three dimensions, ones are implicitly appended to
+    its shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape + y.shape + z.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible objects
+        The three dimensional series is evaluated at the points in the
+        Cartesian product of `x`, `y`, and `z`.  If `x`, `y`, or `z` is a
+        list or tuple, it is first converted to an ndarray, otherwise it is
+        left unchanged and, if it isn't an ndarray, it is treated as a
+        scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree i,j are contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points in the Cartesian
+        product of `x` and `y`.
+
+    See Also
+    --------
+    hermval, hermval2d, hermgrid2d, hermval3d
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermgrid3d
+    >>> x = [1, 2]
+    >>> y = [4, 5]
+    >>> z = [6, 7]
+    >>> c = [[[1, 2, 3], [4, 5, 6]], [[7, 8, 9], [10, 11, 12]]]
+    >>> hermgrid3d(x, y, z, c)
+    array([[[ 40077.,  54117.],
+            [ 49293.,  66561.]],
+           [[ 72375.,  97719.],
+            [ 88975., 120131.]]])
+
+    """
+    return pu._gridnd(hermval, c, x, y, z)
+
+
+def hermvander(x, deg):
+    """Pseudo-Vandermonde matrix of given degree.
+
+    Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
+    `x`. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., i] = H_i(x),
+
+    where ``0 <= i <= deg``. The leading indices of `V` index the elements of
+    `x` and the last index is the degree of the Hermite polynomial.
+
+    If `c` is a 1-D array of coefficients of length ``n + 1`` and `V` is the
+    array ``V = hermvander(x, n)``, then ``np.dot(V, c)`` and
+    ``hermval(x, c)`` are the same up to roundoff. This equivalence is
+    useful both for least squares fitting and for the evaluation of a large
+    number of Hermite series of the same degree and sample points.
+
+    Parameters
+    ----------
+    x : array_like
+        Array of points. The dtype is converted to float64 or complex128
+        depending on whether any of the elements are complex. If `x` is
+        scalar it is converted to a 1-D array.
+    deg : int
+        Degree of the resulting matrix.
+
+    Returns
+    -------
+    vander : ndarray
+        The pseudo-Vandermonde matrix. The shape of the returned matrix is
+        ``x.shape + (deg + 1,)``, where The last index is the degree of the
+        corresponding Hermite polynomial.  The dtype will be the same as
+        the converted `x`.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from numpy.polynomial.hermite import hermvander
+    >>> x = np.array([-1, 0, 1])
+    >>> hermvander(x, 3)
+    array([[ 1., -2.,  2.,  4.],
+           [ 1.,  0., -2., -0.],
+           [ 1.,  2.,  2., -4.]])
+
+    """
+    ideg = pu._as_int(deg, "deg")
+    if ideg < 0:
+        raise ValueError("deg must be non-negative")
+
+    x = np.array(x, copy=None, ndmin=1) + 0.0
+    dims = (ideg + 1,) + x.shape
+    dtyp = x.dtype
+    v = np.empty(dims, dtype=dtyp)
+    v[0] = x * 0 + 1
+    if ideg > 0:
+        x2 = x * 2
+        v[1] = x2
+        for i in range(2, ideg + 1):
+            v[i] = (v[i - 1] * x2 - v[i - 2] * (2 * (i - 1)))
+    return np.moveaxis(v, 0, -1)
+
+
+def hermvander2d(x, y, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points ``(x, y)``. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (deg[1] + 1)*i + j] = H_i(x) * H_j(y),
+
+    where ``0 <= i <= deg[0]`` and ``0 <= j <= deg[1]``. The leading indices of
+    `V` index the points ``(x, y)`` and the last index encodes the degrees of
+    the Hermite polynomials.
+
+    If ``V = hermvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
+    correspond to the elements of a 2-D coefficient array `c` of shape
+    (xdeg + 1, ydeg + 1) in the order
+
+    .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
+
+    and ``np.dot(V, c.flat)`` and ``hermval2d(x, y, c)`` will be the same
+    up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 2-D Hermite
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes
+        will be converted to either float64 or complex128 depending on
+        whether any of the elements are complex. Scalars are converted to 1-D
+        arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg].
+
+    Returns
+    -------
+    vander2d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)`.  The dtype will be the same
+        as the converted `x` and `y`.
+
+    See Also
+    --------
+    hermvander, hermvander3d, hermval2d, hermval3d
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from numpy.polynomial.hermite import hermvander2d
+    >>> x = np.array([-1, 0, 1])
+    >>> y = np.array([-1, 0, 1])
+    >>> hermvander2d(x, y, [2, 2])
+    array([[ 1., -2.,  2., -2.,  4., -4.,  2., -4.,  4.],
+           [ 1.,  0., -2.,  0.,  0., -0., -2., -0.,  4.],
+           [ 1.,  2.,  2.,  2.,  4.,  4.,  2.,  4.,  4.]])
+
+    """
+    return pu._vander_nd_flat((hermvander, hermvander), (x, y), deg)
+
+
+def hermvander3d(x, y, z, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points ``(x, y, z)``. If `l`, `m`, `n` are the given degrees in `x`, `y`, `z`,
+    then The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = H_i(x)*H_j(y)*H_k(z),
+
+    where ``0 <= i <= l``, ``0 <= j <= m``, and ``0 <= j <= n``.  The leading
+    indices of `V` index the points ``(x, y, z)`` and the last index encodes
+    the degrees of the Hermite polynomials.
+
+    If ``V = hermvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
+    of `V` correspond to the elements of a 3-D coefficient array `c` of
+    shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
+
+    .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
+
+    and  ``np.dot(V, c.flat)`` and ``hermval3d(x, y, z, c)`` will be the
+    same up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 3-D Hermite
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y, z : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes will
+        be converted to either float64 or complex128 depending on whether
+        any of the elements are complex. Scalars are converted to 1-D
+        arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg, z_deg].
+
+    Returns
+    -------
+    vander3d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`.  The dtype will
+        be the same as the converted `x`, `y`, and `z`.
+
+    See Also
+    --------
+    hermvander, hermvander3d, hermval2d, hermval3d
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermvander3d
+    >>> x = np.array([-1, 0, 1])
+    >>> y = np.array([-1, 0, 1])
+    >>> z = np.array([-1, 0, 1])
+    >>> hermvander3d(x, y, z, [0, 1, 2])
+    array([[ 1., -2.,  2., -2.,  4., -4.],
+           [ 1.,  0., -2.,  0.,  0., -0.],
+           [ 1.,  2.,  2.,  2.,  4.,  4.]])
+
+    """
+    return pu._vander_nd_flat((hermvander, hermvander, hermvander), (x, y, z), deg)
+
+
+def hermfit(x, y, deg, rcond=None, full=False, w=None):
+    """
+    Least squares fit of Hermite series to data.
+
+    Return the coefficients of a Hermite series of degree `deg` that is the
+    least squares fit to the data values `y` given at points `x`. If `y` is
+    1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
+    fits are done, one for each column of `y`, and the resulting
+    coefficients are stored in the corresponding columns of a 2-D return.
+    The fitted polynomial(s) are in the form
+
+    .. math::  p(x) = c_0 + c_1 * H_1(x) + ... + c_n * H_n(x),
+
+    where `n` is `deg`.
+
+    Parameters
+    ----------
+    x : array_like, shape (M,)
+        x-coordinates of the M sample points ``(x[i], y[i])``.
+    y : array_like, shape (M,) or (M, K)
+        y-coordinates of the sample points. Several data sets of sample
+        points sharing the same x-coordinates can be fitted at once by
+        passing in a 2D-array that contains one dataset per column.
+    deg : int or 1-D array_like
+        Degree(s) of the fitting polynomials. If `deg` is a single integer
+        all terms up to and including the `deg`'th term are included in the
+        fit. For NumPy versions >= 1.11.0 a list of integers specifying the
+        degrees of the terms to include may be used instead.
+    rcond : float, optional
+        Relative condition number of the fit. Singular values smaller than
+        this relative to the largest singular value will be ignored. The
+        default value is len(x)*eps, where eps is the relative precision of
+        the float type, about 2e-16 in most cases.
+    full : bool, optional
+        Switch determining nature of return value. When it is False (the
+        default) just the coefficients are returned, when True diagnostic
+        information from the singular value decomposition is also returned.
+    w : array_like, shape (`M`,), optional
+        Weights. If not None, the weight ``w[i]`` applies to the unsquared
+        residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
+        chosen so that the errors of the products ``w[i]*y[i]`` all have the
+        same variance.  When using inverse-variance weighting, use
+        ``w[i] = 1/sigma(y[i])``.  The default value is None.
+
+    Returns
+    -------
+    coef : ndarray, shape (M,) or (M, K)
+        Hermite coefficients ordered from low to high. If `y` was 2-D,
+        the coefficients for the data in column k  of `y` are in column
+        `k`.
+
+    [residuals, rank, singular_values, rcond] : list
+        These values are only returned if ``full == True``
+
+        - residuals -- sum of squared residuals of the least squares fit
+        - rank -- the numerical rank of the scaled Vandermonde matrix
+        - singular_values -- singular values of the scaled Vandermonde matrix
+        - rcond -- value of `rcond`.
+
+        For more details, see `numpy.linalg.lstsq`.
+
+    Warns
+    -----
+    RankWarning
+        The rank of the coefficient matrix in the least-squares fit is
+        deficient. The warning is only raised if ``full == False``.  The
+        warnings can be turned off by
+
+        >>> import warnings
+        >>> warnings.simplefilter('ignore', np.exceptions.RankWarning)
+
+    See Also
+    --------
+    numpy.polynomial.chebyshev.chebfit
+    numpy.polynomial.legendre.legfit
+    numpy.polynomial.laguerre.lagfit
+    numpy.polynomial.polynomial.polyfit
+    numpy.polynomial.hermite_e.hermefit
+    hermval : Evaluates a Hermite series.
+    hermvander : Vandermonde matrix of Hermite series.
+    hermweight : Hermite weight function
+    numpy.linalg.lstsq : Computes a least-squares fit from the matrix.
+    scipy.interpolate.UnivariateSpline : Computes spline fits.
+
+    Notes
+    -----
+    The solution is the coefficients of the Hermite series `p` that
+    minimizes the sum of the weighted squared errors
+
+    .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
+
+    where the :math:`w_j` are the weights. This problem is solved by
+    setting up the (typically) overdetermined matrix equation
+
+    .. math:: V(x) * c = w * y,
+
+    where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the
+    coefficients to be solved for, `w` are the weights, `y` are the
+    observed values.  This equation is then solved using the singular value
+    decomposition of `V`.
+
+    If some of the singular values of `V` are so small that they are
+    neglected, then a `~exceptions.RankWarning` will be issued. This means that
+    the coefficient values may be poorly determined. Using a lower order fit
+    will usually get rid of the warning.  The `rcond` parameter can also be
+    set to a value smaller than its default, but the resulting fit may be
+    spurious and have large contributions from roundoff error.
+
+    Fits using Hermite series are probably most useful when the data can be
+    approximated by ``sqrt(w(x)) * p(x)``, where ``w(x)`` is the Hermite
+    weight. In that case the weight ``sqrt(w(x[i]))`` should be used
+    together with data values ``y[i]/sqrt(w(x[i]))``. The weight function is
+    available as `hermweight`.
+
+    References
+    ----------
+    .. [1] Wikipedia, "Curve fitting",
+           https://en.wikipedia.org/wiki/Curve_fitting
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from numpy.polynomial.hermite import hermfit, hermval
+    >>> x = np.linspace(-10, 10)
+    >>> rng = np.random.default_rng()
+    >>> err = rng.normal(scale=1./10, size=len(x))
+    >>> y = hermval(x, [1, 2, 3]) + err
+    >>> hermfit(x, y, 2)
+    array([1.02294967, 2.00016403, 2.99994614]) # may vary
+
+    """
+    return pu._fit(hermvander, x, y, deg, rcond, full, w)
+
+
+def hermcompanion(c):
+    """Return the scaled companion matrix of c.
+
+    The basis polynomials are scaled so that the companion matrix is
+    symmetric when `c` is a Hermite basis polynomial. This provides
+    better eigenvalue estimates than the unscaled case and for basis
+    polynomials the eigenvalues are guaranteed to be real if
+    `numpy.linalg.eigvalsh` is used to obtain them.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Hermite series coefficients ordered from low to high
+        degree.
+
+    Returns
+    -------
+    mat : ndarray
+        Scaled companion matrix of dimensions (deg, deg).
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermcompanion
+    >>> hermcompanion([1, 0, 1])
+    array([[0.        , 0.35355339],
+           [0.70710678, 0.        ]])
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) < 2:
+        raise ValueError('Series must have maximum degree of at least 1.')
+    if len(c) == 2:
+        return np.array([[-.5 * c[0] / c[1]]])
+
+    n = len(c) - 1
+    mat = np.zeros((n, n), dtype=c.dtype)
+    scl = np.hstack((1., 1. / np.sqrt(2. * np.arange(n - 1, 0, -1))))
+    scl = np.multiply.accumulate(scl)[::-1]
+    top = mat.reshape(-1)[1::n + 1]
+    bot = mat.reshape(-1)[n::n + 1]
+    top[...] = np.sqrt(.5 * np.arange(1, n))
+    bot[...] = top
+    mat[:, -1] -= scl * c[:-1] / (2.0 * c[-1])
+    return mat
+
+
+def hermroots(c):
+    """
+    Compute the roots of a Hermite series.
+
+    Return the roots (a.k.a. "zeros") of the polynomial
+
+    .. math:: p(x) = \\sum_i c[i] * H_i(x).
+
+    Parameters
+    ----------
+    c : 1-D array_like
+        1-D array of coefficients.
+
+    Returns
+    -------
+    out : ndarray
+        Array of the roots of the series. If all the roots are real,
+        then `out` is also real, otherwise it is complex.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyroots
+    numpy.polynomial.legendre.legroots
+    numpy.polynomial.laguerre.lagroots
+    numpy.polynomial.chebyshev.chebroots
+    numpy.polynomial.hermite_e.hermeroots
+
+    Notes
+    -----
+    The root estimates are obtained as the eigenvalues of the companion
+    matrix, Roots far from the origin of the complex plane may have large
+    errors due to the numerical instability of the series for such
+    values. Roots with multiplicity greater than 1 will also show larger
+    errors as the value of the series near such points is relatively
+    insensitive to errors in the roots. Isolated roots near the origin can
+    be improved by a few iterations of Newton's method.
+
+    The Hermite series basis polynomials aren't powers of `x` so the
+    results of this function may seem unintuitive.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermroots, hermfromroots
+    >>> coef = hermfromroots([-1, 0, 1])
+    >>> coef
+    array([0.   ,  0.25 ,  0.   ,  0.125])
+    >>> hermroots(coef)
+    array([-1.00000000e+00, -1.38777878e-17,  1.00000000e+00])
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) <= 1:
+        return np.array([], dtype=c.dtype)
+    if len(c) == 2:
+        return np.array([-.5 * c[0] / c[1]])
+
+    # rotated companion matrix reduces error
+    m = hermcompanion(c)[::-1, ::-1]
+    r = np.linalg.eigvals(m)
+    r.sort()
+    return r
+
+
+def _normed_hermite_n(x, n):
+    """
+    Evaluate a normalized Hermite polynomial.
+
+    Compute the value of the normalized Hermite polynomial of degree ``n``
+    at the points ``x``.
+
+
+    Parameters
+    ----------
+    x : ndarray of double.
+        Points at which to evaluate the function
+    n : int
+        Degree of the normalized Hermite function to be evaluated.
+
+    Returns
+    -------
+    values : ndarray
+        The shape of the return value is described above.
+
+    Notes
+    -----
+    This function is needed for finding the Gauss points and integration
+    weights for high degrees. The values of the standard Hermite functions
+    overflow when n >= 207.
+
+    """
+    if n == 0:
+        return np.full(x.shape, 1 / np.sqrt(np.sqrt(np.pi)))
+
+    c0 = 0.
+    c1 = 1. / np.sqrt(np.sqrt(np.pi))
+    nd = float(n)
+    for i in range(n - 1):
+        tmp = c0
+        c0 = -c1 * np.sqrt((nd - 1.) / nd)
+        c1 = tmp + c1 * x * np.sqrt(2. / nd)
+        nd = nd - 1.0
+    return c0 + c1 * x * np.sqrt(2)
+
+
+def hermgauss(deg):
+    """
+    Gauss-Hermite quadrature.
+
+    Computes the sample points and weights for Gauss-Hermite quadrature.
+    These sample points and weights will correctly integrate polynomials of
+    degree :math:`2*deg - 1` or less over the interval :math:`[-\\inf, \\inf]`
+    with the weight function :math:`f(x) = \\exp(-x^2)`.
+
+    Parameters
+    ----------
+    deg : int
+        Number of sample points and weights. It must be >= 1.
+
+    Returns
+    -------
+    x : ndarray
+        1-D ndarray containing the sample points.
+    y : ndarray
+        1-D ndarray containing the weights.
+
+    Notes
+    -----
+    The results have only been tested up to degree 100, higher degrees may
+    be problematic. The weights are determined by using the fact that
+
+    .. math:: w_k = c / (H'_n(x_k) * H_{n-1}(x_k))
+
+    where :math:`c` is a constant independent of :math:`k` and :math:`x_k`
+    is the k'th root of :math:`H_n`, and then scaling the results to get
+    the right value when integrating 1.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermgauss
+    >>> hermgauss(2)
+    (array([-0.70710678,  0.70710678]), array([0.88622693, 0.88622693]))
+
+    """
+    ideg = pu._as_int(deg, "deg")
+    if ideg <= 0:
+        raise ValueError("deg must be a positive integer")
+
+    # first approximation of roots. We use the fact that the companion
+    # matrix is symmetric in this case in order to obtain better zeros.
+    c = np.array([0] * deg + [1], dtype=np.float64)
+    m = hermcompanion(c)
+    x = np.linalg.eigvalsh(m)
+
+    # improve roots by one application of Newton
+    dy = _normed_hermite_n(x, ideg)
+    df = _normed_hermite_n(x, ideg - 1) * np.sqrt(2 * ideg)
+    x -= dy / df
+
+    # compute the weights. We scale the factor to avoid possible numerical
+    # overflow.
+    fm = _normed_hermite_n(x, ideg - 1)
+    fm /= np.abs(fm).max()
+    w = 1 / (fm * fm)
+
+    # for Hermite we can also symmetrize
+    w = (w + w[::-1]) / 2
+    x = (x - x[::-1]) / 2
+
+    # scale w to get the right value
+    w *= np.sqrt(np.pi) / w.sum()
+
+    return x, w
+
+
+def hermweight(x):
+    """
+    Weight function of the Hermite polynomials.
+
+    The weight function is :math:`\\exp(-x^2)` and the interval of
+    integration is :math:`[-\\inf, \\inf]`. the Hermite polynomials are
+    orthogonal, but not normalized, with respect to this weight function.
+
+    Parameters
+    ----------
+    x : array_like
+       Values at which the weight function will be computed.
+
+    Returns
+    -------
+    w : ndarray
+       The weight function at `x`.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from numpy.polynomial.hermite import hermweight
+    >>> x = np.arange(-2, 2)
+    >>> hermweight(x)
+    array([0.01831564, 0.36787944, 1.        , 0.36787944])
+
+    """
+    w = np.exp(-x**2)
+    return w
+
+
+#
+# Hermite series class
+#
+
+class Hermite(ABCPolyBase):
+    """A Hermite series class.
+
+    The Hermite class provides the standard Python numerical methods
+    '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the
+    attributes and methods listed below.
+
+    Parameters
+    ----------
+    coef : array_like
+        Hermite coefficients in order of increasing degree, i.e,
+        ``(1, 2, 3)`` gives ``1*H_0(x) + 2*H_1(x) + 3*H_2(x)``.
+    domain : (2,) array_like, optional
+        Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
+        to the interval ``[window[0], window[1]]`` by shifting and scaling.
+        The default value is [-1., 1.].
+    window : (2,) array_like, optional
+        Window, see `domain` for its use. The default value is [-1., 1.].
+    symbol : str, optional
+        Symbol used to represent the independent variable in string
+        representations of the polynomial expression, e.g. for printing.
+        The symbol must be a valid Python identifier. Default value is 'x'.
+
+        .. versionadded:: 1.24
+
+    """
+    # Virtual Functions
+    _add = staticmethod(hermadd)
+    _sub = staticmethod(hermsub)
+    _mul = staticmethod(hermmul)
+    _div = staticmethod(hermdiv)
+    _pow = staticmethod(hermpow)
+    _val = staticmethod(hermval)
+    _int = staticmethod(hermint)
+    _der = staticmethod(hermder)
+    _fit = staticmethod(hermfit)
+    _line = staticmethod(hermline)
+    _roots = staticmethod(hermroots)
+    _fromroots = staticmethod(hermfromroots)
+
+    # Virtual properties
+    domain = np.array(hermdomain)
+    window = np.array(hermdomain)
+    basis_name = 'H'

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy/polynomial/hermite.pyi

@@ -0,0 +1,106 @@
+from typing import Any, ClassVar, Final, Literal as L, TypeVar
+
+import numpy as np
+from numpy._typing import _Shape
+
+from ._polybase import ABCPolyBase
+from ._polytypes import (
+    _Array1,
+    _Array2,
+    _FuncBinOp,
+    _FuncCompanion,
+    _FuncDer,
+    _FuncFit,
+    _FuncFromRoots,
+    _FuncGauss,
+    _FuncInteg,
+    _FuncLine,
+    _FuncPoly2Ortho,
+    _FuncPow,
+    _FuncRoots,
+    _FuncUnOp,
+    _FuncVal,
+    _FuncVal2D,
+    _FuncVal3D,
+    _FuncVander,
+    _FuncVander2D,
+    _FuncVander3D,
+    _FuncWeight,
+)
+from .polyutils import trimcoef as hermtrim
+
+__all__ = [
+    "hermzero",
+    "hermone",
+    "hermx",
+    "hermdomain",
+    "hermline",
+    "hermadd",
+    "hermsub",
+    "hermmulx",
+    "hermmul",
+    "hermdiv",
+    "hermpow",
+    "hermval",
+    "hermder",
+    "hermint",
+    "herm2poly",
+    "poly2herm",
+    "hermfromroots",
+    "hermvander",
+    "hermfit",
+    "hermtrim",
+    "hermroots",
+    "Hermite",
+    "hermval2d",
+    "hermval3d",
+    "hermgrid2d",
+    "hermgrid3d",
+    "hermvander2d",
+    "hermvander3d",
+    "hermcompanion",
+    "hermgauss",
+    "hermweight",
+]
+
+_ShapeT = TypeVar("_ShapeT", bound=_Shape)
+
+poly2herm: Final[_FuncPoly2Ortho] = ...
+herm2poly: Final[_FuncUnOp] = ...
+
+hermdomain: Final[_Array2[np.float64]] = ...
+hermzero: Final[_Array1[np.int_]] = ...
+hermone: Final[_Array1[np.int_]] = ...
+hermx: Final[_Array2[np.int_]] = ...
+
+hermline: Final[_FuncLine] = ...
+hermfromroots: Final[_FuncFromRoots] = ...
+hermadd: Final[_FuncBinOp] = ...
+hermsub: Final[_FuncBinOp] = ...
+hermmulx: Final[_FuncUnOp] = ...
+hermmul: Final[_FuncBinOp] = ...
+hermdiv: Final[_FuncBinOp] = ...
+hermpow: Final[_FuncPow] = ...
+hermder: Final[_FuncDer] = ...
+hermint: Final[_FuncInteg] = ...
+hermval: Final[_FuncVal] = ...
+hermval2d: Final[_FuncVal2D] = ...
+hermval3d: Final[_FuncVal3D] = ...
+hermgrid2d: Final[_FuncVal2D] = ...
+hermgrid3d: Final[_FuncVal3D] = ...
+hermvander: Final[_FuncVander] = ...
+hermvander2d: Final[_FuncVander2D] = ...
+hermvander3d: Final[_FuncVander3D] = ...
+hermfit: Final[_FuncFit] = ...
+hermcompanion: Final[_FuncCompanion] = ...
+hermroots: Final[_FuncRoots] = ...
+
+def _normed_hermite_n(x: np.ndarray[_ShapeT, np.dtype[np.float64]], n: int) -> np.ndarray[_ShapeT, np.dtype[np.float64]]: ...
+
+hermgauss: Final[_FuncGauss] = ...
+hermweight: Final[_FuncWeight] = ...
+
+class Hermite(ABCPolyBase[L["H"]]):
+    basis_name: ClassVar[L["H"]] = "H"  # pyright: ignore[reportIncompatibleMethodOverride]
+    domain: _Array2[np.float64 | Any] = ...  # pyright: ignore[reportIncompatibleMethodOverride]
+    window: _Array2[np.float64 | Any] = ...  # pyright: ignore[reportIncompatibleMethodOverride]

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy/polynomial/hermite_e.py

@@ -0,0 +1,1640 @@
+"""
+===================================================================
+HermiteE Series, "Probabilists" (:mod:`numpy.polynomial.hermite_e`)
+===================================================================
+
+This module provides a number of objects (mostly functions) useful for
+dealing with Hermite_e series, including a `HermiteE` class that
+encapsulates the usual arithmetic operations.  (General information
+on how this module represents and works with such polynomials is in the
+docstring for its "parent" sub-package, `numpy.polynomial`).
+
+Classes
+-------
+.. autosummary::
+   :toctree: generated/
+
+   HermiteE
+
+Constants
+---------
+.. autosummary::
+   :toctree: generated/
+
+   hermedomain
+   hermezero
+   hermeone
+   hermex
+
+Arithmetic
+----------
+.. autosummary::
+   :toctree: generated/
+
+   hermeadd
+   hermesub
+   hermemulx
+   hermemul
+   hermediv
+   hermepow
+   hermeval
+   hermeval2d
+   hermeval3d
+   hermegrid2d
+   hermegrid3d
+
+Calculus
+--------
+.. autosummary::
+   :toctree: generated/
+
+   hermeder
+   hermeint
+
+Misc Functions
+--------------
+.. autosummary::
+   :toctree: generated/
+
+   hermefromroots
+   hermeroots
+   hermevander
+   hermevander2d
+   hermevander3d
+   hermegauss
+   hermeweight
+   hermecompanion
+   hermefit
+   hermetrim
+   hermeline
+   herme2poly
+   poly2herme
+
+See also
+--------
+`numpy.polynomial`
+
+"""
+import numpy as np
+
+from . import polyutils as pu
+from ._polybase import ABCPolyBase
+
+__all__ = [
+    'hermezero', 'hermeone', 'hermex', 'hermedomain', 'hermeline',
+    'hermeadd', 'hermesub', 'hermemulx', 'hermemul', 'hermediv',
+    'hermepow', 'hermeval', 'hermeder', 'hermeint', 'herme2poly',
+    'poly2herme', 'hermefromroots', 'hermevander', 'hermefit', 'hermetrim',
+    'hermeroots', 'HermiteE', 'hermeval2d', 'hermeval3d', 'hermegrid2d',
+    'hermegrid3d', 'hermevander2d', 'hermevander3d', 'hermecompanion',
+    'hermegauss', 'hermeweight']
+
+hermetrim = pu.trimcoef
+
+
+def poly2herme(pol):
+    """
+    poly2herme(pol)
+
+    Convert a polynomial to a Hermite series.
+
+    Convert an array representing the coefficients of a polynomial (relative
+    to the "standard" basis) ordered from lowest degree to highest, to an
+    array of the coefficients of the equivalent Hermite series, ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    pol : array_like
+        1-D array containing the polynomial coefficients
+
+    Returns
+    -------
+    c : ndarray
+        1-D array containing the coefficients of the equivalent Hermite
+        series.
+
+    See Also
+    --------
+    herme2poly
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from numpy.polynomial.hermite_e import poly2herme
+    >>> poly2herme(np.arange(4))
+    array([  2.,  10.,   2.,   3.])
+
+    """
+    [pol] = pu.as_series([pol])
+    deg = len(pol) - 1
+    res = 0
+    for i in range(deg, -1, -1):
+        res = hermeadd(hermemulx(res), pol[i])
+    return res
+
+
+def herme2poly(c):
+    """
+    Convert a Hermite series to a polynomial.
+
+    Convert an array representing the coefficients of a Hermite series,
+    ordered from lowest degree to highest, to an array of the coefficients
+    of the equivalent polynomial (relative to the "standard" basis) ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array containing the Hermite series coefficients, ordered
+        from lowest order term to highest.
+
+    Returns
+    -------
+    pol : ndarray
+        1-D array containing the coefficients of the equivalent polynomial
+        (relative to the "standard" basis) ordered from lowest order term
+        to highest.
+
+    See Also
+    --------
+    poly2herme
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import herme2poly
+    >>> herme2poly([  2.,  10.,   2.,   3.])
+    array([0.,  1.,  2.,  3.])
+
+    """
+    from .polynomial import polyadd, polymulx, polysub
+
+    [c] = pu.as_series([c])
+    n = len(c)
+    if n == 1:
+        return c
+    if n == 2:
+        return c
+    else:
+        c0 = c[-2]
+        c1 = c[-1]
+        # i is the current degree of c1
+        for i in range(n - 1, 1, -1):
+            tmp = c0
+            c0 = polysub(c[i - 2], c1 * (i - 1))
+            c1 = polyadd(tmp, polymulx(c1))
+        return polyadd(c0, polymulx(c1))
+
+
+#
+# These are constant arrays are of integer type so as to be compatible
+# with the widest range of other types, such as Decimal.
+#
+
+# Hermite
+hermedomain = np.array([-1., 1.])
+
+# Hermite coefficients representing zero.
+hermezero = np.array([0])
+
+# Hermite coefficients representing one.
+hermeone = np.array([1])
+
+# Hermite coefficients representing the identity x.
+hermex = np.array([0, 1])
+
+
+def hermeline(off, scl):
+    """
+    Hermite series whose graph is a straight line.
+
+    Parameters
+    ----------
+    off, scl : scalars
+        The specified line is given by ``off + scl*x``.
+
+    Returns
+    -------
+    y : ndarray
+        This module's representation of the Hermite series for
+        ``off + scl*x``.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyline
+    numpy.polynomial.chebyshev.chebline
+    numpy.polynomial.legendre.legline
+    numpy.polynomial.laguerre.lagline
+    numpy.polynomial.hermite.hermline
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermeline
+    >>> from numpy.polynomial.hermite_e import hermeline, hermeval
+    >>> hermeval(0,hermeline(3, 2))
+    3.0
+    >>> hermeval(1,hermeline(3, 2))
+    5.0
+
+    """
+    if scl != 0:
+        return np.array([off, scl])
+    else:
+        return np.array([off])
+
+
+def hermefromroots(roots):
+    """
+    Generate a HermiteE series with given roots.
+
+    The function returns the coefficients of the polynomial
+
+    .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
+
+    in HermiteE form, where the :math:`r_n` are the roots specified in `roots`.
+    If a zero has multiplicity n, then it must appear in `roots` n times.
+    For instance, if 2 is a root of multiplicity three and 3 is a root of
+    multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
+    roots can appear in any order.
+
+    If the returned coefficients are `c`, then
+
+    .. math:: p(x) = c_0 + c_1 * He_1(x) + ... +  c_n * He_n(x)
+
+    The coefficient of the last term is not generally 1 for monic
+    polynomials in HermiteE form.
+
+    Parameters
+    ----------
+    roots : array_like
+        Sequence containing the roots.
+
+    Returns
+    -------
+    out : ndarray
+        1-D array of coefficients.  If all roots are real then `out` is a
+        real array, if some of the roots are complex, then `out` is complex
+        even if all the coefficients in the result are real (see Examples
+        below).
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyfromroots
+    numpy.polynomial.legendre.legfromroots
+    numpy.polynomial.laguerre.lagfromroots
+    numpy.polynomial.hermite.hermfromroots
+    numpy.polynomial.chebyshev.chebfromroots
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermefromroots, hermeval
+    >>> coef = hermefromroots((-1, 0, 1))
+    >>> hermeval((-1, 0, 1), coef)
+    array([0., 0., 0.])
+    >>> coef = hermefromroots((-1j, 1j))
+    >>> hermeval((-1j, 1j), coef)
+    array([0.+0.j, 0.+0.j])
+
+    """
+    return pu._fromroots(hermeline, hermemul, roots)
+
+
+def hermeadd(c1, c2):
+    """
+    Add one Hermite series to another.
+
+    Returns the sum of two Hermite series `c1` + `c2`.  The arguments
+    are sequences of coefficients ordered from lowest order term to
+    highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the Hermite series of their sum.
+
+    See Also
+    --------
+    hermesub, hermemulx, hermemul, hermediv, hermepow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the sum of two Hermite series
+    is a Hermite series (without having to "reproject" the result onto
+    the basis set) so addition, just like that of "standard" polynomials,
+    is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermeadd
+    >>> hermeadd([1, 2, 3], [1, 2, 3, 4])
+    array([2.,  4.,  6.,  4.])
+
+    """
+    return pu._add(c1, c2)
+
+
+def hermesub(c1, c2):
+    """
+    Subtract one Hermite series from another.
+
+    Returns the difference of two Hermite series `c1` - `c2`.  The
+    sequences of coefficients are from lowest order term to highest, i.e.,
+    [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Hermite series coefficients representing their difference.
+
+    See Also
+    --------
+    hermeadd, hermemulx, hermemul, hermediv, hermepow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the difference of two Hermite
+    series is a Hermite series (without having to "reproject" the result
+    onto the basis set) so subtraction, just like that of "standard"
+    polynomials, is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermesub
+    >>> hermesub([1, 2, 3, 4], [1, 2, 3])
+    array([0., 0., 0., 4.])
+
+    """
+    return pu._sub(c1, c2)
+
+
+def hermemulx(c):
+    """Multiply a Hermite series by x.
+
+    Multiply the Hermite series `c` by x, where x is the independent
+    variable.
+
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the result of the multiplication.
+
+    See Also
+    --------
+    hermeadd, hermesub, hermemul, hermediv, hermepow
+
+    Notes
+    -----
+    The multiplication uses the recursion relationship for Hermite
+    polynomials in the form
+
+    .. math::
+
+        xP_i(x) = (P_{i + 1}(x) + iP_{i - 1}(x)))
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermemulx
+    >>> hermemulx([1, 2, 3])
+    array([2.,  7.,  2.,  3.])
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    # The zero series needs special treatment
+    if len(c) == 1 and c[0] == 0:
+        return c
+
+    prd = np.empty(len(c) + 1, dtype=c.dtype)
+    prd[0] = c[0] * 0
+    prd[1] = c[0]
+    for i in range(1, len(c)):
+        prd[i + 1] = c[i]
+        prd[i - 1] += c[i] * i
+    return prd
+
+
+def hermemul(c1, c2):
+    """
+    Multiply one Hermite series by another.
+
+    Returns the product of two Hermite series `c1` * `c2`.  The arguments
+    are sequences of coefficients, from lowest order "term" to highest,
+    e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Hermite series coefficients representing their product.
+
+    See Also
+    --------
+    hermeadd, hermesub, hermemulx, hermediv, hermepow
+
+    Notes
+    -----
+    In general, the (polynomial) product of two C-series results in terms
+    that are not in the Hermite polynomial basis set.  Thus, to express
+    the product as a Hermite series, it is necessary to "reproject" the
+    product onto said basis set, which may produce "unintuitive" (but
+    correct) results; see Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermemul
+    >>> hermemul([1, 2, 3], [0, 1, 2])
+    array([14.,  15.,  28.,   7.,   6.])
+
+    """
+    # s1, s2 are trimmed copies
+    [c1, c2] = pu.as_series([c1, c2])
+
+    if len(c1) > len(c2):
+        c = c2
+        xs = c1
+    else:
+        c = c1
+        xs = c2
+
+    if len(c) == 1:
+        c0 = c[0] * xs
+        c1 = 0
+    elif len(c) == 2:
+        c0 = c[0] * xs
+        c1 = c[1] * xs
+    else:
+        nd = len(c)
+        c0 = c[-2] * xs
+        c1 = c[-1] * xs
+        for i in range(3, len(c) + 1):
+            tmp = c0
+            nd = nd - 1
+            c0 = hermesub(c[-i] * xs, c1 * (nd - 1))
+            c1 = hermeadd(tmp, hermemulx(c1))
+    return hermeadd(c0, hermemulx(c1))
+
+
+def hermediv(c1, c2):
+    """
+    Divide one Hermite series by another.
+
+    Returns the quotient-with-remainder of two Hermite series
+    `c1` / `c2`.  The arguments are sequences of coefficients from lowest
+    order "term" to highest, e.g., [1,2,3] represents the series
+    ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    [quo, rem] : ndarrays
+        Of Hermite series coefficients representing the quotient and
+        remainder.
+
+    See Also
+    --------
+    hermeadd, hermesub, hermemulx, hermemul, hermepow
+
+    Notes
+    -----
+    In general, the (polynomial) division of one Hermite series by another
+    results in quotient and remainder terms that are not in the Hermite
+    polynomial basis set.  Thus, to express these results as a Hermite
+    series, it is necessary to "reproject" the results onto the Hermite
+    basis set, which may produce "unintuitive" (but correct) results; see
+    Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermediv
+    >>> hermediv([ 14.,  15.,  28.,   7.,   6.], [0, 1, 2])
+    (array([1., 2., 3.]), array([0.]))
+    >>> hermediv([ 15.,  17.,  28.,   7.,   6.], [0, 1, 2])
+    (array([1., 2., 3.]), array([1., 2.]))
+
+    """
+    return pu._div(hermemul, c1, c2)
+
+
+def hermepow(c, pow, maxpower=16):
+    """Raise a Hermite series to a power.
+
+    Returns the Hermite series `c` raised to the power `pow`. The
+    argument `c` is a sequence of coefficients ordered from low to high.
+    i.e., [1,2,3] is the series  ``P_0 + 2*P_1 + 3*P_2.``
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Hermite series coefficients ordered from low to
+        high.
+    pow : integer
+        Power to which the series will be raised
+    maxpower : integer, optional
+        Maximum power allowed. This is mainly to limit growth of the series
+        to unmanageable size. Default is 16
+
+    Returns
+    -------
+    coef : ndarray
+        Hermite series of power.
+
+    See Also
+    --------
+    hermeadd, hermesub, hermemulx, hermemul, hermediv
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermepow
+    >>> hermepow([1, 2, 3], 2)
+    array([23.,  28.,  46.,  12.,   9.])
+
+    """
+    return pu._pow(hermemul, c, pow, maxpower)
+
+
+def hermeder(c, m=1, scl=1, axis=0):
+    """
+    Differentiate a Hermite_e series.
+
+    Returns the series coefficients `c` differentiated `m` times along
+    `axis`.  At each iteration the result is multiplied by `scl` (the
+    scaling factor is for use in a linear change of variable). The argument
+    `c` is an array of coefficients from low to high degree along each
+    axis, e.g., [1,2,3] represents the series ``1*He_0 + 2*He_1 + 3*He_2``
+    while [[1,2],[1,2]] represents ``1*He_0(x)*He_0(y) + 1*He_1(x)*He_0(y)
+    + 2*He_0(x)*He_1(y) + 2*He_1(x)*He_1(y)`` if axis=0 is ``x`` and axis=1
+    is ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Hermite_e series coefficients. If `c` is multidimensional
+        the different axis correspond to different variables with the
+        degree in each axis given by the corresponding index.
+    m : int, optional
+        Number of derivatives taken, must be non-negative. (Default: 1)
+    scl : scalar, optional
+        Each differentiation is multiplied by `scl`.  The end result is
+        multiplication by ``scl**m``.  This is for use in a linear change of
+        variable. (Default: 1)
+    axis : int, optional
+        Axis over which the derivative is taken. (Default: 0).
+
+    Returns
+    -------
+    der : ndarray
+        Hermite series of the derivative.
+
+    See Also
+    --------
+    hermeint
+
+    Notes
+    -----
+    In general, the result of differentiating a Hermite series does not
+    resemble the same operation on a power series. Thus the result of this
+    function may be "unintuitive," albeit correct; see Examples section
+    below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermeder
+    >>> hermeder([ 1.,  1.,  1.,  1.])
+    array([1.,  2.,  3.])
+    >>> hermeder([-0.25,  1.,  1./2.,  1./3.,  1./4 ], m=2)
+    array([1.,  2.,  3.])
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    cnt = pu._as_int(m, "the order of derivation")
+    iaxis = pu._as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of derivation must be non-negative")
+    iaxis = np.lib.array_utils.normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    n = len(c)
+    if cnt >= n:
+        return c[:1] * 0
+    else:
+        for i in range(cnt):
+            n = n - 1
+            c *= scl
+            der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
+            for j in range(n, 0, -1):
+                der[j - 1] = j * c[j]
+            c = der
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def hermeint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
+    """
+    Integrate a Hermite_e series.
+
+    Returns the Hermite_e series coefficients `c` integrated `m` times from
+    `lbnd` along `axis`. At each iteration the resulting series is
+    **multiplied** by `scl` and an integration constant, `k`, is added.
+    The scaling factor is for use in a linear change of variable.  ("Buyer
+    beware": note that, depending on what one is doing, one may want `scl`
+    to be the reciprocal of what one might expect; for more information,
+    see the Notes section below.)  The argument `c` is an array of
+    coefficients from low to high degree along each axis, e.g., [1,2,3]
+    represents the series ``H_0 + 2*H_1 + 3*H_2`` while [[1,2],[1,2]]
+    represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) +
+    2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Hermite_e series coefficients. If c is multidimensional
+        the different axis correspond to different variables with the
+        degree in each axis given by the corresponding index.
+    m : int, optional
+        Order of integration, must be positive. (Default: 1)
+    k : {[], list, scalar}, optional
+        Integration constant(s).  The value of the first integral at
+        ``lbnd`` is the first value in the list, the value of the second
+        integral at ``lbnd`` is the second value, etc.  If ``k == []`` (the
+        default), all constants are set to zero.  If ``m == 1``, a single
+        scalar can be given instead of a list.
+    lbnd : scalar, optional
+        The lower bound of the integral. (Default: 0)
+    scl : scalar, optional
+        Following each integration the result is *multiplied* by `scl`
+        before the integration constant is added. (Default: 1)
+    axis : int, optional
+        Axis over which the integral is taken. (Default: 0).
+
+    Returns
+    -------
+    S : ndarray
+        Hermite_e series coefficients of the integral.
+
+    Raises
+    ------
+    ValueError
+        If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
+        ``np.ndim(scl) != 0``.
+
+    See Also
+    --------
+    hermeder
+
+    Notes
+    -----
+    Note that the result of each integration is *multiplied* by `scl`.
+    Why is this important to note?  Say one is making a linear change of
+    variable :math:`u = ax + b` in an integral relative to `x`.  Then
+    :math:`dx = du/a`, so one will need to set `scl` equal to
+    :math:`1/a` - perhaps not what one would have first thought.
+
+    Also note that, in general, the result of integrating a C-series needs
+    to be "reprojected" onto the C-series basis set.  Thus, typically,
+    the result of this function is "unintuitive," albeit correct; see
+    Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermeint
+    >>> hermeint([1, 2, 3]) # integrate once, value 0 at 0.
+    array([1., 1., 1., 1.])
+    >>> hermeint([1, 2, 3], m=2) # integrate twice, value & deriv 0 at 0
+    array([-0.25      ,  1.        ,  0.5       ,  0.33333333,  0.25      ]) # may vary
+    >>> hermeint([1, 2, 3], k=1) # integrate once, value 1 at 0.
+    array([2., 1., 1., 1.])
+    >>> hermeint([1, 2, 3], lbnd=-1) # integrate once, value 0 at -1
+    array([-1.,  1.,  1.,  1.])
+    >>> hermeint([1, 2, 3], m=2, k=[1, 2], lbnd=-1)
+    array([ 1.83333333,  0.        ,  0.5       ,  0.33333333,  0.25      ]) # may vary
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if not np.iterable(k):
+        k = [k]
+    cnt = pu._as_int(m, "the order of integration")
+    iaxis = pu._as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of integration must be non-negative")
+    if len(k) > cnt:
+        raise ValueError("Too many integration constants")
+    if np.ndim(lbnd) != 0:
+        raise ValueError("lbnd must be a scalar.")
+    if np.ndim(scl) != 0:
+        raise ValueError("scl must be a scalar.")
+    iaxis = np.lib.array_utils.normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    k = list(k) + [0] * (cnt - len(k))
+    for i in range(cnt):
+        n = len(c)
+        c *= scl
+        if n == 1 and np.all(c[0] == 0):
+            c[0] += k[i]
+        else:
+            tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype)
+            tmp[0] = c[0] * 0
+            tmp[1] = c[0]
+            for j in range(1, n):
+                tmp[j + 1] = c[j] / (j + 1)
+            tmp[0] += k[i] - hermeval(lbnd, tmp)
+            c = tmp
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def hermeval(x, c, tensor=True):
+    """
+    Evaluate a HermiteE series at points x.
+
+    If `c` is of length ``n + 1``, this function returns the value:
+
+    .. math:: p(x) = c_0 * He_0(x) + c_1 * He_1(x) + ... + c_n * He_n(x)
+
+    The parameter `x` is converted to an array only if it is a tuple or a
+    list, otherwise it is treated as a scalar. In either case, either `x`
+    or its elements must support multiplication and addition both with
+    themselves and with the elements of `c`.
+
+    If `c` is a 1-D array, then ``p(x)`` will have the same shape as `x`.  If
+    `c` is multidimensional, then the shape of the result depends on the
+    value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
+    x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
+    scalars have shape (,).
+
+    Trailing zeros in the coefficients will be used in the evaluation, so
+    they should be avoided if efficiency is a concern.
+
+    Parameters
+    ----------
+    x : array_like, compatible object
+        If `x` is a list or tuple, it is converted to an ndarray, otherwise
+        it is left unchanged and treated as a scalar. In either case, `x`
+        or its elements must support addition and multiplication with
+        with themselves and with the elements of `c`.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree n are contained in c[n]. If `c` is multidimensional the
+        remaining indices enumerate multiple polynomials. In the two
+        dimensional case the coefficients may be thought of as stored in
+        the columns of `c`.
+    tensor : boolean, optional
+        If True, the shape of the coefficient array is extended with ones
+        on the right, one for each dimension of `x`. Scalars have dimension 0
+        for this action. The result is that every column of coefficients in
+        `c` is evaluated for every element of `x`. If False, `x` is broadcast
+        over the columns of `c` for the evaluation.  This keyword is useful
+        when `c` is multidimensional. The default value is True.
+
+    Returns
+    -------
+    values : ndarray, algebra_like
+        The shape of the return value is described above.
+
+    See Also
+    --------
+    hermeval2d, hermegrid2d, hermeval3d, hermegrid3d
+
+    Notes
+    -----
+    The evaluation uses Clenshaw recursion, aka synthetic division.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermeval
+    >>> coef = [1,2,3]
+    >>> hermeval(1, coef)
+    3.0
+    >>> hermeval([[1,2],[3,4]], coef)
+    array([[ 3., 14.],
+           [31., 54.]])
+
+    """
+    c = np.array(c, ndmin=1, copy=None)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if isinstance(x, (tuple, list)):
+        x = np.asarray(x)
+    if isinstance(x, np.ndarray) and tensor:
+        c = c.reshape(c.shape + (1,) * x.ndim)
+
+    if len(c) == 1:
+        c0 = c[0]
+        c1 = 0
+    elif len(c) == 2:
+        c0 = c[0]
+        c1 = c[1]
+    else:
+        nd = len(c)
+        c0 = c[-2]
+        c1 = c[-1]
+        for i in range(3, len(c) + 1):
+            tmp = c0
+            nd = nd - 1
+            c0 = c[-i] - c1 * (nd - 1)
+            c1 = tmp + c1 * x
+    return c0 + c1 * x
+
+
+def hermeval2d(x, y, c):
+    """
+    Evaluate a 2-D HermiteE series at points (x, y).
+
+    This function returns the values:
+
+    .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * He_i(x) * He_j(y)
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars and they
+    must have the same shape after conversion. In either case, either `x`
+    and `y` or their elements must support multiplication and addition both
+    with themselves and with the elements of `c`.
+
+    If `c` is a 1-D array a one is implicitly appended to its shape to make
+    it 2-D. The shape of the result will be c.shape[2:] + x.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points ``(x, y)``,
+        where `x` and `y` must have the same shape. If `x` or `y` is a list
+        or tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and if it isn't an ndarray it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term
+        of multi-degree i,j is contained in ``c[i,j]``. If `c` has
+        dimension greater than two the remaining indices enumerate multiple
+        sets of coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points formed with
+        pairs of corresponding values from `x` and `y`.
+
+    See Also
+    --------
+    hermeval, hermegrid2d, hermeval3d, hermegrid3d
+    """
+    return pu._valnd(hermeval, c, x, y)
+
+
+def hermegrid2d(x, y, c):
+    """
+    Evaluate a 2-D HermiteE series on the Cartesian product of x and y.
+
+    This function returns the values:
+
+    .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * H_i(a) * H_j(b)
+
+    where the points ``(a, b)`` consist of all pairs formed by taking
+    `a` from `x` and `b` from `y`. The resulting points form a grid with
+    `x` in the first dimension and `y` in the second.
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars. In either
+    case, either `x` and `y` or their elements must support multiplication
+    and addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than two dimensions, ones are implicitly appended to
+    its shape to make it 2-D. The shape of the result will be c.shape[2:] +
+    x.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points in the
+        Cartesian product of `x` and `y`.  If `x` or `y` is a list or
+        tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and, if it isn't an ndarray, it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree i,j are contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points in the Cartesian
+        product of `x` and `y`.
+
+    See Also
+    --------
+    hermeval, hermeval2d, hermeval3d, hermegrid3d
+    """
+    return pu._gridnd(hermeval, c, x, y)
+
+
+def hermeval3d(x, y, z, c):
+    """
+    Evaluate a 3-D Hermite_e series at points (x, y, z).
+
+    This function returns the values:
+
+    .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * He_i(x) * He_j(y) * He_k(z)
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if
+    they are tuples or a lists, otherwise they are treated as a scalars and
+    they must have the same shape after conversion. In either case, either
+    `x`, `y`, and `z` or their elements must support multiplication and
+    addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than 3 dimensions, ones are implicitly appended to its
+    shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible object
+        The three dimensional series is evaluated at the points
+        `(x, y, z)`, where `x`, `y`, and `z` must have the same shape.  If
+        any of `x`, `y`, or `z` is a list or tuple, it is first converted
+        to an ndarray, otherwise it is left unchanged and if it isn't an
+        ndarray it is  treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term of
+        multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
+        greater than 3 the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the multidimensional polynomial on points formed with
+        triples of corresponding values from `x`, `y`, and `z`.
+
+    See Also
+    --------
+    hermeval, hermeval2d, hermegrid2d, hermegrid3d
+    """
+    return pu._valnd(hermeval, c, x, y, z)
+
+
+def hermegrid3d(x, y, z, c):
+    """
+    Evaluate a 3-D HermiteE series on the Cartesian product of x, y, and z.
+
+    This function returns the values:
+
+    .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * He_i(a) * He_j(b) * He_k(c)
+
+    where the points ``(a, b, c)`` consist of all triples formed by taking
+    `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
+    a grid with `x` in the first dimension, `y` in the second, and `z` in
+    the third.
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if they
+    are tuples or a lists, otherwise they are treated as a scalars. In
+    either case, either `x`, `y`, and `z` or their elements must support
+    multiplication and addition both with themselves and with the elements
+    of `c`.
+
+    If `c` has fewer than three dimensions, ones are implicitly appended to
+    its shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape + y.shape + z.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible objects
+        The three dimensional series is evaluated at the points in the
+        Cartesian product of `x`, `y`, and `z`.  If `x`, `y`, or `z` is a
+        list or tuple, it is first converted to an ndarray, otherwise it is
+        left unchanged and, if it isn't an ndarray, it is treated as a
+        scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree i,j are contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points in the Cartesian
+        product of `x` and `y`.
+
+    See Also
+    --------
+    hermeval, hermeval2d, hermegrid2d, hermeval3d
+    """
+    return pu._gridnd(hermeval, c, x, y, z)
+
+
+def hermevander(x, deg):
+    """Pseudo-Vandermonde matrix of given degree.
+
+    Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
+    `x`. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., i] = He_i(x),
+
+    where ``0 <= i <= deg``. The leading indices of `V` index the elements of
+    `x` and the last index is the degree of the HermiteE polynomial.
+
+    If `c` is a 1-D array of coefficients of length ``n + 1`` and `V` is the
+    array ``V = hermevander(x, n)``, then ``np.dot(V, c)`` and
+    ``hermeval(x, c)`` are the same up to roundoff. This equivalence is
+    useful both for least squares fitting and for the evaluation of a large
+    number of HermiteE series of the same degree and sample points.
+
+    Parameters
+    ----------
+    x : array_like
+        Array of points. The dtype is converted to float64 or complex128
+        depending on whether any of the elements are complex. If `x` is
+        scalar it is converted to a 1-D array.
+    deg : int
+        Degree of the resulting matrix.
+
+    Returns
+    -------
+    vander : ndarray
+        The pseudo-Vandermonde matrix. The shape of the returned matrix is
+        ``x.shape + (deg + 1,)``, where The last index is the degree of the
+        corresponding HermiteE polynomial.  The dtype will be the same as
+        the converted `x`.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from numpy.polynomial.hermite_e import hermevander
+    >>> x = np.array([-1, 0, 1])
+    >>> hermevander(x, 3)
+    array([[ 1., -1.,  0.,  2.],
+           [ 1.,  0., -1., -0.],
+           [ 1.,  1.,  0., -2.]])
+
+    """
+    ideg = pu._as_int(deg, "deg")
+    if ideg < 0:
+        raise ValueError("deg must be non-negative")
+
+    x = np.array(x, copy=None, ndmin=1) + 0.0
+    dims = (ideg + 1,) + x.shape
+    dtyp = x.dtype
+    v = np.empty(dims, dtype=dtyp)
+    v[0] = x * 0 + 1
+    if ideg > 0:
+        v[1] = x
+        for i in range(2, ideg + 1):
+            v[i] = (v[i - 1] * x - v[i - 2] * (i - 1))
+    return np.moveaxis(v, 0, -1)
+
+
+def hermevander2d(x, y, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points ``(x, y)``. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (deg[1] + 1)*i + j] = He_i(x) * He_j(y),
+
+    where ``0 <= i <= deg[0]`` and ``0 <= j <= deg[1]``. The leading indices of
+    `V` index the points ``(x, y)`` and the last index encodes the degrees of
+    the HermiteE polynomials.
+
+    If ``V = hermevander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
+    correspond to the elements of a 2-D coefficient array `c` of shape
+    (xdeg + 1, ydeg + 1) in the order
+
+    .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
+
+    and ``np.dot(V, c.flat)`` and ``hermeval2d(x, y, c)`` will be the same
+    up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 2-D HermiteE
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes
+        will be converted to either float64 or complex128 depending on
+        whether any of the elements are complex. Scalars are converted to
+        1-D arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg].
+
+    Returns
+    -------
+    vander2d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)`.  The dtype will be the same
+        as the converted `x` and `y`.
+
+    See Also
+    --------
+    hermevander, hermevander3d, hermeval2d, hermeval3d
+    """
+    return pu._vander_nd_flat((hermevander, hermevander), (x, y), deg)
+
+
+def hermevander3d(x, y, z, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points ``(x, y, z)``. If `l`, `m`, `n` are the given degrees in `x`, `y`, `z`,
+    then Hehe pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = He_i(x)*He_j(y)*He_k(z),
+
+    where ``0 <= i <= l``, ``0 <= j <= m``, and ``0 <= j <= n``.  The leading
+    indices of `V` index the points ``(x, y, z)`` and the last index encodes
+    the degrees of the HermiteE polynomials.
+
+    If ``V = hermevander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
+    of `V` correspond to the elements of a 3-D coefficient array `c` of
+    shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
+
+    .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
+
+    and  ``np.dot(V, c.flat)`` and ``hermeval3d(x, y, z, c)`` will be the
+    same up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 3-D HermiteE
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y, z : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes will
+        be converted to either float64 or complex128 depending on whether
+        any of the elements are complex. Scalars are converted to 1-D
+        arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg, z_deg].
+
+    Returns
+    -------
+    vander3d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`.  The dtype will
+        be the same as the converted `x`, `y`, and `z`.
+
+    See Also
+    --------
+    hermevander, hermevander3d, hermeval2d, hermeval3d
+    """
+    return pu._vander_nd_flat((hermevander, hermevander, hermevander), (x, y, z), deg)
+
+
+def hermefit(x, y, deg, rcond=None, full=False, w=None):
+    """
+    Least squares fit of Hermite series to data.
+
+    Return the coefficients of a HermiteE series of degree `deg` that is
+    the least squares fit to the data values `y` given at points `x`. If
+    `y` is 1-D the returned coefficients will also be 1-D. If `y` is 2-D
+    multiple fits are done, one for each column of `y`, and the resulting
+    coefficients are stored in the corresponding columns of a 2-D return.
+    The fitted polynomial(s) are in the form
+
+    .. math::  p(x) = c_0 + c_1 * He_1(x) + ... + c_n * He_n(x),
+
+    where `n` is `deg`.
+
+    Parameters
+    ----------
+    x : array_like, shape (M,)
+        x-coordinates of the M sample points ``(x[i], y[i])``.
+    y : array_like, shape (M,) or (M, K)
+        y-coordinates of the sample points. Several data sets of sample
+        points sharing the same x-coordinates can be fitted at once by
+        passing in a 2D-array that contains one dataset per column.
+    deg : int or 1-D array_like
+        Degree(s) of the fitting polynomials. If `deg` is a single integer
+        all terms up to and including the `deg`'th term are included in the
+        fit. For NumPy versions >= 1.11.0 a list of integers specifying the
+        degrees of the terms to include may be used instead.
+    rcond : float, optional
+        Relative condition number of the fit. Singular values smaller than
+        this relative to the largest singular value will be ignored. The
+        default value is len(x)*eps, where eps is the relative precision of
+        the float type, about 2e-16 in most cases.
+    full : bool, optional
+        Switch determining nature of return value. When it is False (the
+        default) just the coefficients are returned, when True diagnostic
+        information from the singular value decomposition is also returned.
+    w : array_like, shape (`M`,), optional
+        Weights. If not None, the weight ``w[i]`` applies to the unsquared
+        residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
+        chosen so that the errors of the products ``w[i]*y[i]`` all have the
+        same variance.  When using inverse-variance weighting, use
+        ``w[i] = 1/sigma(y[i])``.  The default value is None.
+
+    Returns
+    -------
+    coef : ndarray, shape (M,) or (M, K)
+        Hermite coefficients ordered from low to high. If `y` was 2-D,
+        the coefficients for the data in column k  of `y` are in column
+        `k`.
+
+    [residuals, rank, singular_values, rcond] : list
+        These values are only returned if ``full == True``
+
+        - residuals -- sum of squared residuals of the least squares fit
+        - rank -- the numerical rank of the scaled Vandermonde matrix
+        - singular_values -- singular values of the scaled Vandermonde matrix
+        - rcond -- value of `rcond`.
+
+        For more details, see `numpy.linalg.lstsq`.
+
+    Warns
+    -----
+    RankWarning
+        The rank of the coefficient matrix in the least-squares fit is
+        deficient. The warning is only raised if ``full = False``.  The
+        warnings can be turned off by
+
+        >>> import warnings
+        >>> warnings.simplefilter('ignore', np.exceptions.RankWarning)
+
+    See Also
+    --------
+    numpy.polynomial.chebyshev.chebfit
+    numpy.polynomial.legendre.legfit
+    numpy.polynomial.polynomial.polyfit
+    numpy.polynomial.hermite.hermfit
+    numpy.polynomial.laguerre.lagfit
+    hermeval : Evaluates a Hermite series.
+    hermevander : pseudo Vandermonde matrix of Hermite series.
+    hermeweight : HermiteE weight function.
+    numpy.linalg.lstsq : Computes a least-squares fit from the matrix.
+    scipy.interpolate.UnivariateSpline : Computes spline fits.
+
+    Notes
+    -----
+    The solution is the coefficients of the HermiteE series `p` that
+    minimizes the sum of the weighted squared errors
+
+    .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
+
+    where the :math:`w_j` are the weights. This problem is solved by
+    setting up the (typically) overdetermined matrix equation
+
+    .. math:: V(x) * c = w * y,
+
+    where `V` is the pseudo Vandermonde matrix of `x`, the elements of `c`
+    are the coefficients to be solved for, and the elements of `y` are the
+    observed values.  This equation is then solved using the singular value
+    decomposition of `V`.
+
+    If some of the singular values of `V` are so small that they are
+    neglected, then a `~exceptions.RankWarning` will be issued. This means that
+    the coefficient values may be poorly determined. Using a lower order fit
+    will usually get rid of the warning.  The `rcond` parameter can also be
+    set to a value smaller than its default, but the resulting fit may be
+    spurious and have large contributions from roundoff error.
+
+    Fits using HermiteE series are probably most useful when the data can
+    be approximated by ``sqrt(w(x)) * p(x)``, where ``w(x)`` is the HermiteE
+    weight. In that case the weight ``sqrt(w(x[i]))`` should be used
+    together with data values ``y[i]/sqrt(w(x[i]))``. The weight function is
+    available as `hermeweight`.
+
+    References
+    ----------
+    .. [1] Wikipedia, "Curve fitting",
+           https://en.wikipedia.org/wiki/Curve_fitting
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from numpy.polynomial.hermite_e import hermefit, hermeval
+    >>> x = np.linspace(-10, 10)
+    >>> rng = np.random.default_rng()
+    >>> err = rng.normal(scale=1./10, size=len(x))
+    >>> y = hermeval(x, [1, 2, 3]) + err
+    >>> hermefit(x, y, 2)
+    array([1.02284196, 2.00032805, 2.99978457]) # may vary
+
+    """
+    return pu._fit(hermevander, x, y, deg, rcond, full, w)
+
+
+def hermecompanion(c):
+    """
+    Return the scaled companion matrix of c.
+
+    The basis polynomials are scaled so that the companion matrix is
+    symmetric when `c` is a HermiteE basis polynomial. This provides
+    better eigenvalue estimates than the unscaled case and for basis
+    polynomials the eigenvalues are guaranteed to be real if
+    `numpy.linalg.eigvalsh` is used to obtain them.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of HermiteE series coefficients ordered from low to high
+        degree.
+
+    Returns
+    -------
+    mat : ndarray
+        Scaled companion matrix of dimensions (deg, deg).
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) < 2:
+        raise ValueError('Series must have maximum degree of at least 1.')
+    if len(c) == 2:
+        return np.array([[-c[0] / c[1]]])
+
+    n = len(c) - 1
+    mat = np.zeros((n, n), dtype=c.dtype)
+    scl = np.hstack((1., 1. / np.sqrt(np.arange(n - 1, 0, -1))))
+    scl = np.multiply.accumulate(scl)[::-1]
+    top = mat.reshape(-1)[1::n + 1]
+    bot = mat.reshape(-1)[n::n + 1]
+    top[...] = np.sqrt(np.arange(1, n))
+    bot[...] = top
+    mat[:, -1] -= scl * c[:-1] / c[-1]
+    return mat
+
+
+def hermeroots(c):
+    """
+    Compute the roots of a HermiteE series.
+
+    Return the roots (a.k.a. "zeros") of the polynomial
+
+    .. math:: p(x) = \\sum_i c[i] * He_i(x).
+
+    Parameters
+    ----------
+    c : 1-D array_like
+        1-D array of coefficients.
+
+    Returns
+    -------
+    out : ndarray
+        Array of the roots of the series. If all the roots are real,
+        then `out` is also real, otherwise it is complex.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyroots
+    numpy.polynomial.legendre.legroots
+    numpy.polynomial.laguerre.lagroots
+    numpy.polynomial.hermite.hermroots
+    numpy.polynomial.chebyshev.chebroots
+
+    Notes
+    -----
+    The root estimates are obtained as the eigenvalues of the companion
+    matrix, Roots far from the origin of the complex plane may have large
+    errors due to the numerical instability of the series for such
+    values. Roots with multiplicity greater than 1 will also show larger
+    errors as the value of the series near such points is relatively
+    insensitive to errors in the roots. Isolated roots near the origin can
+    be improved by a few iterations of Newton's method.
+
+    The HermiteE series basis polynomials aren't powers of `x` so the
+    results of this function may seem unintuitive.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermeroots, hermefromroots
+    >>> coef = hermefromroots([-1, 0, 1])
+    >>> coef
+    array([0., 2., 0., 1.])
+    >>> hermeroots(coef)
+    array([-1.,  0.,  1.]) # may vary
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) <= 1:
+        return np.array([], dtype=c.dtype)
+    if len(c) == 2:
+        return np.array([-c[0] / c[1]])
+
+    # rotated companion matrix reduces error
+    m = hermecompanion(c)[::-1, ::-1]
+    r = np.linalg.eigvals(m)
+    r.sort()
+    return r
+
+
+def _normed_hermite_e_n(x, n):
+    """
+    Evaluate a normalized HermiteE polynomial.
+
+    Compute the value of the normalized HermiteE polynomial of degree ``n``
+    at the points ``x``.
+
+
+    Parameters
+    ----------
+    x : ndarray of double.
+        Points at which to evaluate the function
+    n : int
+        Degree of the normalized HermiteE function to be evaluated.
+
+    Returns
+    -------
+    values : ndarray
+        The shape of the return value is described above.
+
+    Notes
+    -----
+    This function is needed for finding the Gauss points and integration
+    weights for high degrees. The values of the standard HermiteE functions
+    overflow when n >= 207.
+
+    """
+    if n == 0:
+        return np.full(x.shape, 1 / np.sqrt(np.sqrt(2 * np.pi)))
+
+    c0 = 0.
+    c1 = 1. / np.sqrt(np.sqrt(2 * np.pi))
+    nd = float(n)
+    for i in range(n - 1):
+        tmp = c0
+        c0 = -c1 * np.sqrt((nd - 1.) / nd)
+        c1 = tmp + c1 * x * np.sqrt(1. / nd)
+        nd = nd - 1.0
+    return c0 + c1 * x
+
+
+def hermegauss(deg):
+    """
+    Gauss-HermiteE quadrature.
+
+    Computes the sample points and weights for Gauss-HermiteE quadrature.
+    These sample points and weights will correctly integrate polynomials of
+    degree :math:`2*deg - 1` or less over the interval :math:`[-\\inf, \\inf]`
+    with the weight function :math:`f(x) = \\exp(-x^2/2)`.
+
+    Parameters
+    ----------
+    deg : int
+        Number of sample points and weights. It must be >= 1.
+
+    Returns
+    -------
+    x : ndarray
+        1-D ndarray containing the sample points.
+    y : ndarray
+        1-D ndarray containing the weights.
+
+    Notes
+    -----
+    The results have only been tested up to degree 100, higher degrees may
+    be problematic. The weights are determined by using the fact that
+
+    .. math:: w_k = c / (He'_n(x_k) * He_{n-1}(x_k))
+
+    where :math:`c` is a constant independent of :math:`k` and :math:`x_k`
+    is the k'th root of :math:`He_n`, and then scaling the results to get
+    the right value when integrating 1.
+
+    """
+    ideg = pu._as_int(deg, "deg")
+    if ideg <= 0:
+        raise ValueError("deg must be a positive integer")
+
+    # first approximation of roots. We use the fact that the companion
+    # matrix is symmetric in this case in order to obtain better zeros.
+    c = np.array([0] * deg + [1])
+    m = hermecompanion(c)
+    x = np.linalg.eigvalsh(m)
+
+    # improve roots by one application of Newton
+    dy = _normed_hermite_e_n(x, ideg)
+    df = _normed_hermite_e_n(x, ideg - 1) * np.sqrt(ideg)
+    x -= dy / df
+
+    # compute the weights. We scale the factor to avoid possible numerical
+    # overflow.
+    fm = _normed_hermite_e_n(x, ideg - 1)
+    fm /= np.abs(fm).max()
+    w = 1 / (fm * fm)
+
+    # for Hermite_e we can also symmetrize
+    w = (w + w[::-1]) / 2
+    x = (x - x[::-1]) / 2
+
+    # scale w to get the right value
+    w *= np.sqrt(2 * np.pi) / w.sum()
+
+    return x, w
+
+
+def hermeweight(x):
+    """Weight function of the Hermite_e polynomials.
+
+    The weight function is :math:`\\exp(-x^2/2)` and the interval of
+    integration is :math:`[-\\inf, \\inf]`. the HermiteE polynomials are
+    orthogonal, but not normalized, with respect to this weight function.
+
+    Parameters
+    ----------
+    x : array_like
+       Values at which the weight function will be computed.
+
+    Returns
+    -------
+    w : ndarray
+       The weight function at `x`.
+    """
+    w = np.exp(-.5 * x**2)
+    return w
+
+
+#
+# HermiteE series class
+#
+
+class HermiteE(ABCPolyBase):
+    """A HermiteE series class.
+
+    The HermiteE class provides the standard Python numerical methods
+    '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the
+    attributes and methods listed below.
+
+    Parameters
+    ----------
+    coef : array_like
+        HermiteE coefficients in order of increasing degree, i.e,
+        ``(1, 2, 3)`` gives ``1*He_0(x) + 2*He_1(X) + 3*He_2(x)``.
+    domain : (2,) array_like, optional
+        Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
+        to the interval ``[window[0], window[1]]`` by shifting and scaling.
+        The default value is [-1., 1.].
+    window : (2,) array_like, optional
+        Window, see `domain` for its use. The default value is [-1., 1.].
+    symbol : str, optional
+        Symbol used to represent the independent variable in string
+        representations of the polynomial expression, e.g. for printing.
+        The symbol must be a valid Python identifier. Default value is 'x'.
+
+        .. versionadded:: 1.24
+
+    """
+    # Virtual Functions
+    _add = staticmethod(hermeadd)
+    _sub = staticmethod(hermesub)
+    _mul = staticmethod(hermemul)
+    _div = staticmethod(hermediv)
+    _pow = staticmethod(hermepow)
+    _val = staticmethod(hermeval)
+    _int = staticmethod(hermeint)
+    _der = staticmethod(hermeder)
+    _fit = staticmethod(hermefit)
+    _line = staticmethod(hermeline)
+    _roots = staticmethod(hermeroots)
+    _fromroots = staticmethod(hermefromroots)
+
+    # Virtual properties
+    domain = np.array(hermedomain)
+    window = np.array(hermedomain)
+    basis_name = 'He'

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy/polynomial/hermite_e.pyi

@@ -0,0 +1,106 @@
+from typing import Any, ClassVar, Final, Literal as L, TypeVar
+
+import numpy as np
+from numpy._typing import _Shape
+
+from ._polybase import ABCPolyBase
+from ._polytypes import (
+    _Array1,
+    _Array2,
+    _FuncBinOp,
+    _FuncCompanion,
+    _FuncDer,
+    _FuncFit,
+    _FuncFromRoots,
+    _FuncGauss,
+    _FuncInteg,
+    _FuncLine,
+    _FuncPoly2Ortho,
+    _FuncPow,
+    _FuncRoots,
+    _FuncUnOp,
+    _FuncVal,
+    _FuncVal2D,
+    _FuncVal3D,
+    _FuncVander,
+    _FuncVander2D,
+    _FuncVander3D,
+    _FuncWeight,
+)
+from .polyutils import trimcoef as hermetrim
+
+__all__ = [
+    "hermezero",
+    "hermeone",
+    "hermex",
+    "hermedomain",
+    "hermeline",
+    "hermeadd",
+    "hermesub",
+    "hermemulx",
+    "hermemul",
+    "hermediv",
+    "hermepow",
+    "hermeval",
+    "hermeder",
+    "hermeint",
+    "herme2poly",
+    "poly2herme",
+    "hermefromroots",
+    "hermevander",
+    "hermefit",
+    "hermetrim",
+    "hermeroots",
+    "HermiteE",
+    "hermeval2d",
+    "hermeval3d",
+    "hermegrid2d",
+    "hermegrid3d",
+    "hermevander2d",
+    "hermevander3d",
+    "hermecompanion",
+    "hermegauss",
+    "hermeweight",
+]
+
+_ShapeT = TypeVar("_ShapeT", bound=_Shape)
+
+poly2herme: Final[_FuncPoly2Ortho] = ...
+herme2poly: Final[_FuncUnOp] = ...
+
+hermedomain: Final[_Array2[np.float64]] = ...
+hermezero: Final[_Array1[np.int_]] = ...
+hermeone: Final[_Array1[np.int_]] = ...
+hermex: Final[_Array2[np.int_]] = ...
+
+hermeline: Final[_FuncLine] = ...
+hermefromroots: Final[_FuncFromRoots] = ...
+hermeadd: Final[_FuncBinOp] = ...
+hermesub: Final[_FuncBinOp] = ...
+hermemulx: Final[_FuncUnOp] = ...
+hermemul: Final[_FuncBinOp] = ...
+hermediv: Final[_FuncBinOp] = ...
+hermepow: Final[_FuncPow] = ...
+hermeder: Final[_FuncDer] = ...
+hermeint: Final[_FuncInteg] = ...
+hermeval: Final[_FuncVal] = ...
+hermeval2d: Final[_FuncVal2D] = ...
+hermeval3d: Final[_FuncVal3D] = ...
+hermegrid2d: Final[_FuncVal2D] = ...
+hermegrid3d: Final[_FuncVal3D] = ...
+hermevander: Final[_FuncVander] = ...
+hermevander2d: Final[_FuncVander2D] = ...
+hermevander3d: Final[_FuncVander3D] = ...
+hermefit: Final[_FuncFit] = ...
+hermecompanion: Final[_FuncCompanion] = ...
+hermeroots: Final[_FuncRoots] = ...
+
+def _normed_hermite_e_n(x: np.ndarray[_ShapeT, np.dtype[np.float64]], n: int) -> np.ndarray[_ShapeT, np.dtype[np.float64]]: ...
+
+hermegauss: Final[_FuncGauss] = ...
+hermeweight: Final[_FuncWeight] = ...
+
+class HermiteE(ABCPolyBase[L["He"]]):
+    basis_name: ClassVar[L["He"]] = "He"  # pyright: ignore[reportIncompatibleMethodOverride]
+    domain: _Array2[np.float64 | Any] = ...  # pyright: ignore[reportIncompatibleMethodOverride]
+    window: _Array2[np.float64 | Any] = ...  # pyright: ignore[reportIncompatibleMethodOverride]

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy/polynomial/laguerre.py

@@ -0,0 +1,1673 @@
+"""
+==================================================
+Laguerre Series (:mod:`numpy.polynomial.laguerre`)
+==================================================
+
+This module provides a number of objects (mostly functions) useful for
+dealing with Laguerre series, including a `Laguerre` class that
+encapsulates the usual arithmetic operations.  (General information
+on how this module represents and works with such polynomials is in the
+docstring for its "parent" sub-package, `numpy.polynomial`).
+
+Classes
+-------
+.. autosummary::
+   :toctree: generated/
+
+   Laguerre
+
+Constants
+---------
+.. autosummary::
+   :toctree: generated/
+
+   lagdomain
+   lagzero
+   lagone
+   lagx
+
+Arithmetic
+----------
+.. autosummary::
+   :toctree: generated/
+
+   lagadd
+   lagsub
+   lagmulx
+   lagmul
+   lagdiv
+   lagpow
+   lagval
+   lagval2d
+   lagval3d
+   laggrid2d
+   laggrid3d
+
+Calculus
+--------
+.. autosummary::
+   :toctree: generated/
+
+   lagder
+   lagint
+
+Misc Functions
+--------------
+.. autosummary::
+   :toctree: generated/
+
+   lagfromroots
+   lagroots
+   lagvander
+   lagvander2d
+   lagvander3d
+   laggauss
+   lagweight
+   lagcompanion
+   lagfit
+   lagtrim
+   lagline
+   lag2poly
+   poly2lag
+
+See also
+--------
+`numpy.polynomial`
+
+"""
+import numpy as np
+
+from . import polyutils as pu
+from ._polybase import ABCPolyBase
+
+__all__ = [
+    'lagzero', 'lagone', 'lagx', 'lagdomain', 'lagline', 'lagadd',
+    'lagsub', 'lagmulx', 'lagmul', 'lagdiv', 'lagpow', 'lagval', 'lagder',
+    'lagint', 'lag2poly', 'poly2lag', 'lagfromroots', 'lagvander',
+    'lagfit', 'lagtrim', 'lagroots', 'Laguerre', 'lagval2d', 'lagval3d',
+    'laggrid2d', 'laggrid3d', 'lagvander2d', 'lagvander3d', 'lagcompanion',
+    'laggauss', 'lagweight']
+
+lagtrim = pu.trimcoef
+
+
+def poly2lag(pol):
+    """
+    poly2lag(pol)
+
+    Convert a polynomial to a Laguerre series.
+
+    Convert an array representing the coefficients of a polynomial (relative
+    to the "standard" basis) ordered from lowest degree to highest, to an
+    array of the coefficients of the equivalent Laguerre series, ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    pol : array_like
+        1-D array containing the polynomial coefficients
+
+    Returns
+    -------
+    c : ndarray
+        1-D array containing the coefficients of the equivalent Laguerre
+        series.
+
+    See Also
+    --------
+    lag2poly
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from numpy.polynomial.laguerre import poly2lag
+    >>> poly2lag(np.arange(4))
+    array([ 23., -63.,  58., -18.])
+
+    """
+    [pol] = pu.as_series([pol])
+    res = 0
+    for p in pol[::-1]:
+        res = lagadd(lagmulx(res), p)
+    return res
+
+
+def lag2poly(c):
+    """
+    Convert a Laguerre series to a polynomial.
+
+    Convert an array representing the coefficients of a Laguerre series,
+    ordered from lowest degree to highest, to an array of the coefficients
+    of the equivalent polynomial (relative to the "standard" basis) ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array containing the Laguerre series coefficients, ordered
+        from lowest order term to highest.
+
+    Returns
+    -------
+    pol : ndarray
+        1-D array containing the coefficients of the equivalent polynomial
+        (relative to the "standard" basis) ordered from lowest order term
+        to highest.
+
+    See Also
+    --------
+    poly2lag
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lag2poly
+    >>> lag2poly([ 23., -63.,  58., -18.])
+    array([0., 1., 2., 3.])
+
+    """
+    from .polynomial import polyadd, polymulx, polysub
+
+    [c] = pu.as_series([c])
+    n = len(c)
+    if n == 1:
+        return c
+    else:
+        c0 = c[-2]
+        c1 = c[-1]
+        # i is the current degree of c1
+        for i in range(n - 1, 1, -1):
+            tmp = c0
+            c0 = polysub(c[i - 2], (c1 * (i - 1)) / i)
+            c1 = polyadd(tmp, polysub((2 * i - 1) * c1, polymulx(c1)) / i)
+        return polyadd(c0, polysub(c1, polymulx(c1)))
+
+
+#
+# These are constant arrays are of integer type so as to be compatible
+# with the widest range of other types, such as Decimal.
+#
+
+# Laguerre
+lagdomain = np.array([0., 1.])
+
+# Laguerre coefficients representing zero.
+lagzero = np.array([0])
+
+# Laguerre coefficients representing one.
+lagone = np.array([1])
+
+# Laguerre coefficients representing the identity x.
+lagx = np.array([1, -1])
+
+
+def lagline(off, scl):
+    """
+    Laguerre series whose graph is a straight line.
+
+    Parameters
+    ----------
+    off, scl : scalars
+        The specified line is given by ``off + scl*x``.
+
+    Returns
+    -------
+    y : ndarray
+        This module's representation of the Laguerre series for
+        ``off + scl*x``.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyline
+    numpy.polynomial.chebyshev.chebline
+    numpy.polynomial.legendre.legline
+    numpy.polynomial.hermite.hermline
+    numpy.polynomial.hermite_e.hermeline
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagline, lagval
+    >>> lagval(0,lagline(3, 2))
+    3.0
+    >>> lagval(1,lagline(3, 2))
+    5.0
+
+    """
+    if scl != 0:
+        return np.array([off + scl, -scl])
+    else:
+        return np.array([off])
+
+
+def lagfromroots(roots):
+    """
+    Generate a Laguerre series with given roots.
+
+    The function returns the coefficients of the polynomial
+
+    .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
+
+    in Laguerre form, where the :math:`r_n` are the roots specified in `roots`.
+    If a zero has multiplicity n, then it must appear in `roots` n times.
+    For instance, if 2 is a root of multiplicity three and 3 is a root of
+    multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
+    roots can appear in any order.
+
+    If the returned coefficients are `c`, then
+
+    .. math:: p(x) = c_0 + c_1 * L_1(x) + ... +  c_n * L_n(x)
+
+    The coefficient of the last term is not generally 1 for monic
+    polynomials in Laguerre form.
+
+    Parameters
+    ----------
+    roots : array_like
+        Sequence containing the roots.
+
+    Returns
+    -------
+    out : ndarray
+        1-D array of coefficients.  If all roots are real then `out` is a
+        real array, if some of the roots are complex, then `out` is complex
+        even if all the coefficients in the result are real (see Examples
+        below).
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyfromroots
+    numpy.polynomial.legendre.legfromroots
+    numpy.polynomial.chebyshev.chebfromroots
+    numpy.polynomial.hermite.hermfromroots
+    numpy.polynomial.hermite_e.hermefromroots
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagfromroots, lagval
+    >>> coef = lagfromroots((-1, 0, 1))
+    >>> lagval((-1, 0, 1), coef)
+    array([0.,  0.,  0.])
+    >>> coef = lagfromroots((-1j, 1j))
+    >>> lagval((-1j, 1j), coef)
+    array([0.+0.j, 0.+0.j])
+
+    """
+    return pu._fromroots(lagline, lagmul, roots)
+
+
+def lagadd(c1, c2):
+    """
+    Add one Laguerre series to another.
+
+    Returns the sum of two Laguerre series `c1` + `c2`.  The arguments
+    are sequences of coefficients ordered from lowest order term to
+    highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Laguerre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the Laguerre series of their sum.
+
+    See Also
+    --------
+    lagsub, lagmulx, lagmul, lagdiv, lagpow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the sum of two Laguerre series
+    is a Laguerre series (without having to "reproject" the result onto
+    the basis set) so addition, just like that of "standard" polynomials,
+    is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagadd
+    >>> lagadd([1, 2, 3], [1, 2, 3, 4])
+    array([2.,  4.,  6.,  4.])
+
+    """
+    return pu._add(c1, c2)
+
+
+def lagsub(c1, c2):
+    """
+    Subtract one Laguerre series from another.
+
+    Returns the difference of two Laguerre series `c1` - `c2`.  The
+    sequences of coefficients are from lowest order term to highest, i.e.,
+    [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Laguerre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Laguerre series coefficients representing their difference.
+
+    See Also
+    --------
+    lagadd, lagmulx, lagmul, lagdiv, lagpow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the difference of two Laguerre
+    series is a Laguerre series (without having to "reproject" the result
+    onto the basis set) so subtraction, just like that of "standard"
+    polynomials, is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagsub
+    >>> lagsub([1, 2, 3, 4], [1, 2, 3])
+    array([0.,  0.,  0.,  4.])
+
+    """
+    return pu._sub(c1, c2)
+
+
+def lagmulx(c):
+    """Multiply a Laguerre series by x.
+
+    Multiply the Laguerre series `c` by x, where x is the independent
+    variable.
+
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Laguerre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the result of the multiplication.
+
+    See Also
+    --------
+    lagadd, lagsub, lagmul, lagdiv, lagpow
+
+    Notes
+    -----
+    The multiplication uses the recursion relationship for Laguerre
+    polynomials in the form
+
+    .. math::
+
+        xP_i(x) = (-(i + 1)*P_{i + 1}(x) + (2i + 1)P_{i}(x) - iP_{i - 1}(x))
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagmulx
+    >>> lagmulx([1, 2, 3])
+    array([-1.,  -1.,  11.,  -9.])
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    # The zero series needs special treatment
+    if len(c) == 1 and c[0] == 0:
+        return c
+
+    prd = np.empty(len(c) + 1, dtype=c.dtype)
+    prd[0] = c[0]
+    prd[1] = -c[0]
+    for i in range(1, len(c)):
+        prd[i + 1] = -c[i] * (i + 1)
+        prd[i] += c[i] * (2 * i + 1)
+        prd[i - 1] -= c[i] * i
+    return prd
+
+
+def lagmul(c1, c2):
+    """
+    Multiply one Laguerre series by another.
+
+    Returns the product of two Laguerre series `c1` * `c2`.  The arguments
+    are sequences of coefficients, from lowest order "term" to highest,
+    e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Laguerre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Laguerre series coefficients representing their product.
+
+    See Also
+    --------
+    lagadd, lagsub, lagmulx, lagdiv, lagpow
+
+    Notes
+    -----
+    In general, the (polynomial) product of two C-series results in terms
+    that are not in the Laguerre polynomial basis set.  Thus, to express
+    the product as a Laguerre series, it is necessary to "reproject" the
+    product onto said basis set, which may produce "unintuitive" (but
+    correct) results; see Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagmul
+    >>> lagmul([1, 2, 3], [0, 1, 2])
+    array([  8., -13.,  38., -51.,  36.])
+
+    """
+    # s1, s2 are trimmed copies
+    [c1, c2] = pu.as_series([c1, c2])
+
+    if len(c1) > len(c2):
+        c = c2
+        xs = c1
+    else:
+        c = c1
+        xs = c2
+
+    if len(c) == 1:
+        c0 = c[0] * xs
+        c1 = 0
+    elif len(c) == 2:
+        c0 = c[0] * xs
+        c1 = c[1] * xs
+    else:
+        nd = len(c)
+        c0 = c[-2] * xs
+        c1 = c[-1] * xs
+        for i in range(3, len(c) + 1):
+            tmp = c0
+            nd = nd - 1
+            c0 = lagsub(c[-i] * xs, (c1 * (nd - 1)) / nd)
+            c1 = lagadd(tmp, lagsub((2 * nd - 1) * c1, lagmulx(c1)) / nd)
+    return lagadd(c0, lagsub(c1, lagmulx(c1)))
+
+
+def lagdiv(c1, c2):
+    """
+    Divide one Laguerre series by another.
+
+    Returns the quotient-with-remainder of two Laguerre series
+    `c1` / `c2`.  The arguments are sequences of coefficients from lowest
+    order "term" to highest, e.g., [1,2,3] represents the series
+    ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Laguerre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    [quo, rem] : ndarrays
+        Of Laguerre series coefficients representing the quotient and
+        remainder.
+
+    See Also
+    --------
+    lagadd, lagsub, lagmulx, lagmul, lagpow
+
+    Notes
+    -----
+    In general, the (polynomial) division of one Laguerre series by another
+    results in quotient and remainder terms that are not in the Laguerre
+    polynomial basis set.  Thus, to express these results as a Laguerre
+    series, it is necessary to "reproject" the results onto the Laguerre
+    basis set, which may produce "unintuitive" (but correct) results; see
+    Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagdiv
+    >>> lagdiv([  8., -13.,  38., -51.,  36.], [0, 1, 2])
+    (array([1., 2., 3.]), array([0.]))
+    >>> lagdiv([  9., -12.,  38., -51.,  36.], [0, 1, 2])
+    (array([1., 2., 3.]), array([1., 1.]))
+
+    """
+    return pu._div(lagmul, c1, c2)
+
+
+def lagpow(c, pow, maxpower=16):
+    """Raise a Laguerre series to a power.
+
+    Returns the Laguerre series `c` raised to the power `pow`. The
+    argument `c` is a sequence of coefficients ordered from low to high.
+    i.e., [1,2,3] is the series  ``P_0 + 2*P_1 + 3*P_2.``
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Laguerre series coefficients ordered from low to
+        high.
+    pow : integer
+        Power to which the series will be raised
+    maxpower : integer, optional
+        Maximum power allowed. This is mainly to limit growth of the series
+        to unmanageable size. Default is 16
+
+    Returns
+    -------
+    coef : ndarray
+        Laguerre series of power.
+
+    See Also
+    --------
+    lagadd, lagsub, lagmulx, lagmul, lagdiv
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagpow
+    >>> lagpow([1, 2, 3], 2)
+    array([ 14., -16.,  56., -72.,  54.])
+
+    """
+    return pu._pow(lagmul, c, pow, maxpower)
+
+
+def lagder(c, m=1, scl=1, axis=0):
+    """
+    Differentiate a Laguerre series.
+
+    Returns the Laguerre series coefficients `c` differentiated `m` times
+    along `axis`.  At each iteration the result is multiplied by `scl` (the
+    scaling factor is for use in a linear change of variable). The argument
+    `c` is an array of coefficients from low to high degree along each
+    axis, e.g., [1,2,3] represents the series ``1*L_0 + 2*L_1 + 3*L_2``
+    while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) +
+    2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is
+    ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Laguerre series coefficients. If `c` is multidimensional
+        the different axis correspond to different variables with the
+        degree in each axis given by the corresponding index.
+    m : int, optional
+        Number of derivatives taken, must be non-negative. (Default: 1)
+    scl : scalar, optional
+        Each differentiation is multiplied by `scl`.  The end result is
+        multiplication by ``scl**m``.  This is for use in a linear change of
+        variable. (Default: 1)
+    axis : int, optional
+        Axis over which the derivative is taken. (Default: 0).
+
+    Returns
+    -------
+    der : ndarray
+        Laguerre series of the derivative.
+
+    See Also
+    --------
+    lagint
+
+    Notes
+    -----
+    In general, the result of differentiating a Laguerre series does not
+    resemble the same operation on a power series. Thus the result of this
+    function may be "unintuitive," albeit correct; see Examples section
+    below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagder
+    >>> lagder([ 1.,  1.,  1., -3.])
+    array([1.,  2.,  3.])
+    >>> lagder([ 1.,  0.,  0., -4.,  3.], m=2)
+    array([1.,  2.,  3.])
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+
+    cnt = pu._as_int(m, "the order of derivation")
+    iaxis = pu._as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of derivation must be non-negative")
+    iaxis = np.lib.array_utils.normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    n = len(c)
+    if cnt >= n:
+        c = c[:1] * 0
+    else:
+        for i in range(cnt):
+            n = n - 1
+            c *= scl
+            der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
+            for j in range(n, 1, -1):
+                der[j - 1] = -c[j]
+                c[j - 1] += c[j]
+            der[0] = -c[1]
+            c = der
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def lagint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
+    """
+    Integrate a Laguerre series.
+
+    Returns the Laguerre series coefficients `c` integrated `m` times from
+    `lbnd` along `axis`. At each iteration the resulting series is
+    **multiplied** by `scl` and an integration constant, `k`, is added.
+    The scaling factor is for use in a linear change of variable.  ("Buyer
+    beware": note that, depending on what one is doing, one may want `scl`
+    to be the reciprocal of what one might expect; for more information,
+    see the Notes section below.)  The argument `c` is an array of
+    coefficients from low to high degree along each axis, e.g., [1,2,3]
+    represents the series ``L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]]
+    represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) +
+    2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
+
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Laguerre series coefficients. If `c` is multidimensional
+        the different axis correspond to different variables with the
+        degree in each axis given by the corresponding index.
+    m : int, optional
+        Order of integration, must be positive. (Default: 1)
+    k : {[], list, scalar}, optional
+        Integration constant(s).  The value of the first integral at
+        ``lbnd`` is the first value in the list, the value of the second
+        integral at ``lbnd`` is the second value, etc.  If ``k == []`` (the
+        default), all constants are set to zero.  If ``m == 1``, a single
+        scalar can be given instead of a list.
+    lbnd : scalar, optional
+        The lower bound of the integral. (Default: 0)
+    scl : scalar, optional
+        Following each integration the result is *multiplied* by `scl`
+        before the integration constant is added. (Default: 1)
+    axis : int, optional
+        Axis over which the integral is taken. (Default: 0).
+
+    Returns
+    -------
+    S : ndarray
+        Laguerre series coefficients of the integral.
+
+    Raises
+    ------
+    ValueError
+        If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
+        ``np.ndim(scl) != 0``.
+
+    See Also
+    --------
+    lagder
+
+    Notes
+    -----
+    Note that the result of each integration is *multiplied* by `scl`.
+    Why is this important to note?  Say one is making a linear change of
+    variable :math:`u = ax + b` in an integral relative to `x`.  Then
+    :math:`dx = du/a`, so one will need to set `scl` equal to
+    :math:`1/a` - perhaps not what one would have first thought.
+
+    Also note that, in general, the result of integrating a C-series needs
+    to be "reprojected" onto the C-series basis set.  Thus, typically,
+    the result of this function is "unintuitive," albeit correct; see
+    Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagint
+    >>> lagint([1,2,3])
+    array([ 1.,  1.,  1., -3.])
+    >>> lagint([1,2,3], m=2)
+    array([ 1.,  0.,  0., -4.,  3.])
+    >>> lagint([1,2,3], k=1)
+    array([ 2.,  1.,  1., -3.])
+    >>> lagint([1,2,3], lbnd=-1)
+    array([11.5,  1. ,  1. , -3. ])
+    >>> lagint([1,2], m=2, k=[1,2], lbnd=-1)
+    array([ 11.16666667,  -5.        ,  -3.        ,   2.        ]) # may vary
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if not np.iterable(k):
+        k = [k]
+    cnt = pu._as_int(m, "the order of integration")
+    iaxis = pu._as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of integration must be non-negative")
+    if len(k) > cnt:
+        raise ValueError("Too many integration constants")
+    if np.ndim(lbnd) != 0:
+        raise ValueError("lbnd must be a scalar.")
+    if np.ndim(scl) != 0:
+        raise ValueError("scl must be a scalar.")
+    iaxis = np.lib.array_utils.normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    k = list(k) + [0] * (cnt - len(k))
+    for i in range(cnt):
+        n = len(c)
+        c *= scl
+        if n == 1 and np.all(c[0] == 0):
+            c[0] += k[i]
+        else:
+            tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype)
+            tmp[0] = c[0]
+            tmp[1] = -c[0]
+            for j in range(1, n):
+                tmp[j] += c[j]
+                tmp[j + 1] = -c[j]
+            tmp[0] += k[i] - lagval(lbnd, tmp)
+            c = tmp
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def lagval(x, c, tensor=True):
+    """
+    Evaluate a Laguerre series at points x.
+
+    If `c` is of length ``n + 1``, this function returns the value:
+
+    .. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)
+
+    The parameter `x` is converted to an array only if it is a tuple or a
+    list, otherwise it is treated as a scalar. In either case, either `x`
+    or its elements must support multiplication and addition both with
+    themselves and with the elements of `c`.
+
+    If `c` is a 1-D array, then ``p(x)`` will have the same shape as `x`.  If
+    `c` is multidimensional, then the shape of the result depends on the
+    value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
+    x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
+    scalars have shape (,).
+
+    Trailing zeros in the coefficients will be used in the evaluation, so
+    they should be avoided if efficiency is a concern.
+
+    Parameters
+    ----------
+    x : array_like, compatible object
+        If `x` is a list or tuple, it is converted to an ndarray, otherwise
+        it is left unchanged and treated as a scalar. In either case, `x`
+        or its elements must support addition and multiplication with
+        themselves and with the elements of `c`.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree n are contained in c[n]. If `c` is multidimensional the
+        remaining indices enumerate multiple polynomials. In the two
+        dimensional case the coefficients may be thought of as stored in
+        the columns of `c`.
+    tensor : boolean, optional
+        If True, the shape of the coefficient array is extended with ones
+        on the right, one for each dimension of `x`. Scalars have dimension 0
+        for this action. The result is that every column of coefficients in
+        `c` is evaluated for every element of `x`. If False, `x` is broadcast
+        over the columns of `c` for the evaluation.  This keyword is useful
+        when `c` is multidimensional. The default value is True.
+
+    Returns
+    -------
+    values : ndarray, algebra_like
+        The shape of the return value is described above.
+
+    See Also
+    --------
+    lagval2d, laggrid2d, lagval3d, laggrid3d
+
+    Notes
+    -----
+    The evaluation uses Clenshaw recursion, aka synthetic division.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagval
+    >>> coef = [1, 2, 3]
+    >>> lagval(1, coef)
+    -0.5
+    >>> lagval([[1, 2],[3, 4]], coef)
+    array([[-0.5, -4. ],
+           [-4.5, -2. ]])
+
+    """
+    c = np.array(c, ndmin=1, copy=None)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if isinstance(x, (tuple, list)):
+        x = np.asarray(x)
+    if isinstance(x, np.ndarray) and tensor:
+        c = c.reshape(c.shape + (1,) * x.ndim)
+
+    if len(c) == 1:
+        c0 = c[0]
+        c1 = 0
+    elif len(c) == 2:
+        c0 = c[0]
+        c1 = c[1]
+    else:
+        nd = len(c)
+        c0 = c[-2]
+        c1 = c[-1]
+        for i in range(3, len(c) + 1):
+            tmp = c0
+            nd = nd - 1
+            c0 = c[-i] - (c1 * (nd - 1)) / nd
+            c1 = tmp + (c1 * ((2 * nd - 1) - x)) / nd
+    return c0 + c1 * (1 - x)
+
+
+def lagval2d(x, y, c):
+    """
+    Evaluate a 2-D Laguerre series at points (x, y).
+
+    This function returns the values:
+
+    .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_j(y)
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars and they
+    must have the same shape after conversion. In either case, either `x`
+    and `y` or their elements must support multiplication and addition both
+    with themselves and with the elements of `c`.
+
+    If `c` is a 1-D array a one is implicitly appended to its shape to make
+    it 2-D. The shape of the result will be c.shape[2:] + x.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points ``(x, y)``,
+        where `x` and `y` must have the same shape. If `x` or `y` is a list
+        or tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and if it isn't an ndarray it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term
+        of multi-degree i,j is contained in ``c[i,j]``. If `c` has
+        dimension greater than two the remaining indices enumerate multiple
+        sets of coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points formed with
+        pairs of corresponding values from `x` and `y`.
+
+    See Also
+    --------
+    lagval, laggrid2d, lagval3d, laggrid3d
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagval2d
+    >>> c = [[1, 2],[3, 4]]
+    >>> lagval2d(1, 1, c)
+    1.0
+    """
+    return pu._valnd(lagval, c, x, y)
+
+
+def laggrid2d(x, y, c):
+    """
+    Evaluate a 2-D Laguerre series on the Cartesian product of x and y.
+
+    This function returns the values:
+
+    .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * L_i(a) * L_j(b)
+
+    where the points ``(a, b)`` consist of all pairs formed by taking
+    `a` from `x` and `b` from `y`. The resulting points form a grid with
+    `x` in the first dimension and `y` in the second.
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars. In either
+    case, either `x` and `y` or their elements must support multiplication
+    and addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than two dimensions, ones are implicitly appended to
+    its shape to make it 2-D. The shape of the result will be c.shape[2:] +
+    x.shape + y.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points in the
+        Cartesian product of `x` and `y`.  If `x` or `y` is a list or
+        tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and, if it isn't an ndarray, it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term of
+        multi-degree i,j is contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional Chebyshev series at points in the
+        Cartesian product of `x` and `y`.
+
+    See Also
+    --------
+    lagval, lagval2d, lagval3d, laggrid3d
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import laggrid2d
+    >>> c = [[1, 2], [3, 4]]
+    >>> laggrid2d([0, 1], [0, 1], c)
+    array([[10.,  4.],
+           [ 3.,  1.]])
+
+    """
+    return pu._gridnd(lagval, c, x, y)
+
+
+def lagval3d(x, y, z, c):
+    """
+    Evaluate a 3-D Laguerre series at points (x, y, z).
+
+    This function returns the values:
+
+    .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_k(z)
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if
+    they are tuples or a lists, otherwise they are treated as a scalars and
+    they must have the same shape after conversion. In either case, either
+    `x`, `y`, and `z` or their elements must support multiplication and
+    addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than 3 dimensions, ones are implicitly appended to its
+    shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible object
+        The three dimensional series is evaluated at the points
+        ``(x, y, z)``, where `x`, `y`, and `z` must have the same shape.  If
+        any of `x`, `y`, or `z` is a list or tuple, it is first converted
+        to an ndarray, otherwise it is left unchanged and if it isn't an
+        ndarray it is  treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term of
+        multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
+        greater than 3 the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the multidimensional polynomial on points formed with
+        triples of corresponding values from `x`, `y`, and `z`.
+
+    See Also
+    --------
+    lagval, lagval2d, laggrid2d, laggrid3d
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagval3d
+    >>> c = [[[1, 2], [3, 4]], [[5, 6], [7, 8]]]
+    >>> lagval3d(1, 1, 2, c)
+    -1.0
+
+    """
+    return pu._valnd(lagval, c, x, y, z)
+
+
+def laggrid3d(x, y, z, c):
+    """
+    Evaluate a 3-D Laguerre series on the Cartesian product of x, y, and z.
+
+    This function returns the values:
+
+    .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * L_i(a) * L_j(b) * L_k(c)
+
+    where the points ``(a, b, c)`` consist of all triples formed by taking
+    `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
+    a grid with `x` in the first dimension, `y` in the second, and `z` in
+    the third.
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if they
+    are tuples or a lists, otherwise they are treated as a scalars. In
+    either case, either `x`, `y`, and `z` or their elements must support
+    multiplication and addition both with themselves and with the elements
+    of `c`.
+
+    If `c` has fewer than three dimensions, ones are implicitly appended to
+    its shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape + y.shape + z.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible objects
+        The three dimensional series is evaluated at the points in the
+        Cartesian product of `x`, `y`, and `z`.  If `x`, `y`, or `z` is a
+        list or tuple, it is first converted to an ndarray, otherwise it is
+        left unchanged and, if it isn't an ndarray, it is treated as a
+        scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree i,j are contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points in the Cartesian
+        product of `x` and `y`.
+
+    See Also
+    --------
+    lagval, lagval2d, laggrid2d, lagval3d
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import laggrid3d
+    >>> c = [[[1, 2], [3, 4]], [[5, 6], [7, 8]]]
+    >>> laggrid3d([0, 1], [0, 1], [2, 4], c)
+    array([[[ -4., -44.],
+            [ -2., -18.]],
+           [[ -2., -14.],
+            [ -1.,  -5.]]])
+
+    """
+    return pu._gridnd(lagval, c, x, y, z)
+
+
+def lagvander(x, deg):
+    """Pseudo-Vandermonde matrix of given degree.
+
+    Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
+    `x`. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., i] = L_i(x)
+
+    where ``0 <= i <= deg``. The leading indices of `V` index the elements of
+    `x` and the last index is the degree of the Laguerre polynomial.
+
+    If `c` is a 1-D array of coefficients of length ``n + 1`` and `V` is the
+    array ``V = lagvander(x, n)``, then ``np.dot(V, c)`` and
+    ``lagval(x, c)`` are the same up to roundoff. This equivalence is
+    useful both for least squares fitting and for the evaluation of a large
+    number of Laguerre series of the same degree and sample points.
+
+    Parameters
+    ----------
+    x : array_like
+        Array of points. The dtype is converted to float64 or complex128
+        depending on whether any of the elements are complex. If `x` is
+        scalar it is converted to a 1-D array.
+    deg : int
+        Degree of the resulting matrix.
+
+    Returns
+    -------
+    vander : ndarray
+        The pseudo-Vandermonde matrix. The shape of the returned matrix is
+        ``x.shape + (deg + 1,)``, where The last index is the degree of the
+        corresponding Laguerre polynomial.  The dtype will be the same as
+        the converted `x`.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from numpy.polynomial.laguerre import lagvander
+    >>> x = np.array([0, 1, 2])
+    >>> lagvander(x, 3)
+    array([[ 1.        ,  1.        ,  1.        ,  1.        ],
+           [ 1.        ,  0.        , -0.5       , -0.66666667],
+           [ 1.        , -1.        , -1.        , -0.33333333]])
+
+    """
+    ideg = pu._as_int(deg, "deg")
+    if ideg < 0:
+        raise ValueError("deg must be non-negative")
+
+    x = np.array(x, copy=None, ndmin=1) + 0.0
+    dims = (ideg + 1,) + x.shape
+    dtyp = x.dtype
+    v = np.empty(dims, dtype=dtyp)
+    v[0] = x * 0 + 1
+    if ideg > 0:
+        v[1] = 1 - x
+        for i in range(2, ideg + 1):
+            v[i] = (v[i - 1] * (2 * i - 1 - x) - v[i - 2] * (i - 1)) / i
+    return np.moveaxis(v, 0, -1)
+
+
+def lagvander2d(x, y, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points ``(x, y)``. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (deg[1] + 1)*i + j] = L_i(x) * L_j(y),
+
+    where ``0 <= i <= deg[0]`` and ``0 <= j <= deg[1]``. The leading indices of
+    `V` index the points ``(x, y)`` and the last index encodes the degrees of
+    the Laguerre polynomials.
+
+    If ``V = lagvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
+    correspond to the elements of a 2-D coefficient array `c` of shape
+    (xdeg + 1, ydeg + 1) in the order
+
+    .. math:: c_{00}, c_{01}, c_{02}, ... , c_{10}, c_{11}, c_{12}, ...
+
+    and ``np.dot(V, c.flat)`` and ``lagval2d(x, y, c)`` will be the same
+    up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 2-D Laguerre
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes
+        will be converted to either float64 or complex128 depending on
+        whether any of the elements are complex. Scalars are converted to
+        1-D arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg].
+
+    Returns
+    -------
+    vander2d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)`.  The dtype will be the same
+        as the converted `x` and `y`.
+
+    See Also
+    --------
+    lagvander, lagvander3d, lagval2d, lagval3d
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from numpy.polynomial.laguerre import lagvander2d
+    >>> x = np.array([0])
+    >>> y = np.array([2])
+    >>> lagvander2d(x, y, [2, 1])
+    array([[ 1., -1.,  1., -1.,  1., -1.]])
+
+    """
+    return pu._vander_nd_flat((lagvander, lagvander), (x, y), deg)
+
+
+def lagvander3d(x, y, z, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points ``(x, y, z)``. If `l`, `m`, `n` are the given degrees in `x`, `y`, `z`,
+    then The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = L_i(x)*L_j(y)*L_k(z),
+
+    where ``0 <= i <= l``, ``0 <= j <= m``, and ``0 <= j <= n``.  The leading
+    indices of `V` index the points ``(x, y, z)`` and the last index encodes
+    the degrees of the Laguerre polynomials.
+
+    If ``V = lagvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
+    of `V` correspond to the elements of a 3-D coefficient array `c` of
+    shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
+
+    .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
+
+    and  ``np.dot(V, c.flat)`` and ``lagval3d(x, y, z, c)`` will be the
+    same up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 3-D Laguerre
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y, z : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes will
+        be converted to either float64 or complex128 depending on whether
+        any of the elements are complex. Scalars are converted to 1-D
+        arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg, z_deg].
+
+    Returns
+    -------
+    vander3d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`.  The dtype will
+        be the same as the converted `x`, `y`, and `z`.
+
+    See Also
+    --------
+    lagvander, lagvander3d, lagval2d, lagval3d
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from numpy.polynomial.laguerre import lagvander3d
+    >>> x = np.array([0])
+    >>> y = np.array([2])
+    >>> z = np.array([0])
+    >>> lagvander3d(x, y, z, [2, 1, 3])
+    array([[ 1.,  1.,  1.,  1., -1., -1., -1., -1.,  1.,  1.,  1.,  1., -1.,
+            -1., -1., -1.,  1.,  1.,  1.,  1., -1., -1., -1., -1.]])
+
+    """
+    return pu._vander_nd_flat((lagvander, lagvander, lagvander), (x, y, z), deg)
+
+
+def lagfit(x, y, deg, rcond=None, full=False, w=None):
+    """
+    Least squares fit of Laguerre series to data.
+
+    Return the coefficients of a Laguerre series of degree `deg` that is the
+    least squares fit to the data values `y` given at points `x`. If `y` is
+    1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
+    fits are done, one for each column of `y`, and the resulting
+    coefficients are stored in the corresponding columns of a 2-D return.
+    The fitted polynomial(s) are in the form
+
+    .. math::  p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x),
+
+    where ``n`` is `deg`.
+
+    Parameters
+    ----------
+    x : array_like, shape (M,)
+        x-coordinates of the M sample points ``(x[i], y[i])``.
+    y : array_like, shape (M,) or (M, K)
+        y-coordinates of the sample points. Several data sets of sample
+        points sharing the same x-coordinates can be fitted at once by
+        passing in a 2D-array that contains one dataset per column.
+    deg : int or 1-D array_like
+        Degree(s) of the fitting polynomials. If `deg` is a single integer
+        all terms up to and including the `deg`'th term are included in the
+        fit. For NumPy versions >= 1.11.0 a list of integers specifying the
+        degrees of the terms to include may be used instead.
+    rcond : float, optional
+        Relative condition number of the fit. Singular values smaller than
+        this relative to the largest singular value will be ignored. The
+        default value is len(x)*eps, where eps is the relative precision of
+        the float type, about 2e-16 in most cases.
+    full : bool, optional
+        Switch determining nature of return value. When it is False (the
+        default) just the coefficients are returned, when True diagnostic
+        information from the singular value decomposition is also returned.
+    w : array_like, shape (`M`,), optional
+        Weights. If not None, the weight ``w[i]`` applies to the unsquared
+        residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
+        chosen so that the errors of the products ``w[i]*y[i]`` all have the
+        same variance.  When using inverse-variance weighting, use
+        ``w[i] = 1/sigma(y[i])``.  The default value is None.
+
+    Returns
+    -------
+    coef : ndarray, shape (M,) or (M, K)
+        Laguerre coefficients ordered from low to high. If `y` was 2-D,
+        the coefficients for the data in column *k*  of `y` are in column
+        *k*.
+
+    [residuals, rank, singular_values, rcond] : list
+        These values are only returned if ``full == True``
+
+        - residuals -- sum of squared residuals of the least squares fit
+        - rank -- the numerical rank of the scaled Vandermonde matrix
+        - singular_values -- singular values of the scaled Vandermonde matrix
+        - rcond -- value of `rcond`.
+
+        For more details, see `numpy.linalg.lstsq`.
+
+    Warns
+    -----
+    RankWarning
+        The rank of the coefficient matrix in the least-squares fit is
+        deficient. The warning is only raised if ``full == False``.  The
+        warnings can be turned off by
+
+        >>> import warnings
+        >>> warnings.simplefilter('ignore', np.exceptions.RankWarning)
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyfit
+    numpy.polynomial.legendre.legfit
+    numpy.polynomial.chebyshev.chebfit
+    numpy.polynomial.hermite.hermfit
+    numpy.polynomial.hermite_e.hermefit
+    lagval : Evaluates a Laguerre series.
+    lagvander : pseudo Vandermonde matrix of Laguerre series.
+    lagweight : Laguerre weight function.
+    numpy.linalg.lstsq : Computes a least-squares fit from the matrix.
+    scipy.interpolate.UnivariateSpline : Computes spline fits.
+
+    Notes
+    -----
+    The solution is the coefficients of the Laguerre series ``p`` that
+    minimizes the sum of the weighted squared errors
+
+    .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
+
+    where the :math:`w_j` are the weights. This problem is solved by
+    setting up as the (typically) overdetermined matrix equation
+
+    .. math:: V(x) * c = w * y,
+
+    where ``V`` is the weighted pseudo Vandermonde matrix of `x`, ``c`` are the
+    coefficients to be solved for, `w` are the weights, and `y` are the
+    observed values.  This equation is then solved using the singular value
+    decomposition of ``V``.
+
+    If some of the singular values of `V` are so small that they are
+    neglected, then a `~exceptions.RankWarning` will be issued. This means that
+    the coefficient values may be poorly determined. Using a lower order fit
+    will usually get rid of the warning.  The `rcond` parameter can also be
+    set to a value smaller than its default, but the resulting fit may be
+    spurious and have large contributions from roundoff error.
+
+    Fits using Laguerre series are probably most useful when the data can
+    be approximated by ``sqrt(w(x)) * p(x)``, where ``w(x)`` is the Laguerre
+    weight. In that case the weight ``sqrt(w(x[i]))`` should be used
+    together with data values ``y[i]/sqrt(w(x[i]))``. The weight function is
+    available as `lagweight`.
+
+    References
+    ----------
+    .. [1] Wikipedia, "Curve fitting",
+           https://en.wikipedia.org/wiki/Curve_fitting
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from numpy.polynomial.laguerre import lagfit, lagval
+    >>> x = np.linspace(0, 10)
+    >>> rng = np.random.default_rng()
+    >>> err = rng.normal(scale=1./10, size=len(x))
+    >>> y = lagval(x, [1, 2, 3]) + err
+    >>> lagfit(x, y, 2)
+    array([1.00578369, 1.99417356, 2.99827656]) # may vary
+
+    """
+    return pu._fit(lagvander, x, y, deg, rcond, full, w)
+
+
+def lagcompanion(c):
+    """
+    Return the companion matrix of c.
+
+    The usual companion matrix of the Laguerre polynomials is already
+    symmetric when `c` is a basis Laguerre polynomial, so no scaling is
+    applied.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Laguerre series coefficients ordered from low to high
+        degree.
+
+    Returns
+    -------
+    mat : ndarray
+        Companion matrix of dimensions (deg, deg).
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagcompanion
+    >>> lagcompanion([1, 2, 3])
+    array([[ 1.        , -0.33333333],
+           [-1.        ,  4.33333333]])
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) < 2:
+        raise ValueError('Series must have maximum degree of at least 1.')
+    if len(c) == 2:
+        return np.array([[1 + c[0] / c[1]]])
+
+    n = len(c) - 1
+    mat = np.zeros((n, n), dtype=c.dtype)
+    top = mat.reshape(-1)[1::n + 1]
+    mid = mat.reshape(-1)[0::n + 1]
+    bot = mat.reshape(-1)[n::n + 1]
+    top[...] = -np.arange(1, n)
+    mid[...] = 2. * np.arange(n) + 1.
+    bot[...] = top
+    mat[:, -1] += (c[:-1] / c[-1]) * n
+    return mat
+
+
+def lagroots(c):
+    """
+    Compute the roots of a Laguerre series.
+
+    Return the roots (a.k.a. "zeros") of the polynomial
+
+    .. math:: p(x) = \\sum_i c[i] * L_i(x).
+
+    Parameters
+    ----------
+    c : 1-D array_like
+        1-D array of coefficients.
+
+    Returns
+    -------
+    out : ndarray
+        Array of the roots of the series. If all the roots are real,
+        then `out` is also real, otherwise it is complex.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyroots
+    numpy.polynomial.legendre.legroots
+    numpy.polynomial.chebyshev.chebroots
+    numpy.polynomial.hermite.hermroots
+    numpy.polynomial.hermite_e.hermeroots
+
+    Notes
+    -----
+    The root estimates are obtained as the eigenvalues of the companion
+    matrix, Roots far from the origin of the complex plane may have large
+    errors due to the numerical instability of the series for such
+    values. Roots with multiplicity greater than 1 will also show larger
+    errors as the value of the series near such points is relatively
+    insensitive to errors in the roots. Isolated roots near the origin can
+    be improved by a few iterations of Newton's method.
+
+    The Laguerre series basis polynomials aren't powers of `x` so the
+    results of this function may seem unintuitive.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagroots, lagfromroots
+    >>> coef = lagfromroots([0, 1, 2])
+    >>> coef
+    array([  2.,  -8.,  12.,  -6.])
+    >>> lagroots(coef)
+    array([-4.4408921e-16,  1.0000000e+00,  2.0000000e+00])
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) <= 1:
+        return np.array([], dtype=c.dtype)
+    if len(c) == 2:
+        return np.array([1 + c[0] / c[1]])
+
+    # rotated companion matrix reduces error
+    m = lagcompanion(c)[::-1, ::-1]
+    r = np.linalg.eigvals(m)
+    r.sort()
+    return r
+
+
+def laggauss(deg):
+    """
+    Gauss-Laguerre quadrature.
+
+    Computes the sample points and weights for Gauss-Laguerre quadrature.
+    These sample points and weights will correctly integrate polynomials of
+    degree :math:`2*deg - 1` or less over the interval :math:`[0, \\inf]`
+    with the weight function :math:`f(x) = \\exp(-x)`.
+
+    Parameters
+    ----------
+    deg : int
+        Number of sample points and weights. It must be >= 1.
+
+    Returns
+    -------
+    x : ndarray
+        1-D ndarray containing the sample points.
+    y : ndarray
+        1-D ndarray containing the weights.
+
+    Notes
+    -----
+    The results have only been tested up to degree 100 higher degrees may
+    be problematic. The weights are determined by using the fact that
+
+    .. math:: w_k = c / (L'_n(x_k) * L_{n-1}(x_k))
+
+    where :math:`c` is a constant independent of :math:`k` and :math:`x_k`
+    is the k'th root of :math:`L_n`, and then scaling the results to get
+    the right value when integrating 1.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import laggauss
+    >>> laggauss(2)
+    (array([0.58578644, 3.41421356]), array([0.85355339, 0.14644661]))
+
+    """
+    ideg = pu._as_int(deg, "deg")
+    if ideg <= 0:
+        raise ValueError("deg must be a positive integer")
+
+    # first approximation of roots. We use the fact that the companion
+    # matrix is symmetric in this case in order to obtain better zeros.
+    c = np.array([0] * deg + [1])
+    m = lagcompanion(c)
+    x = np.linalg.eigvalsh(m)
+
+    # improve roots by one application of Newton
+    dy = lagval(x, c)
+    df = lagval(x, lagder(c))
+    x -= dy / df
+
+    # compute the weights. We scale the factor to avoid possible numerical
+    # overflow.
+    fm = lagval(x, c[1:])
+    fm /= np.abs(fm).max()
+    df /= np.abs(df).max()
+    w = 1 / (fm * df)
+
+    # scale w to get the right value, 1 in this case
+    w /= w.sum()
+
+    return x, w
+
+
+def lagweight(x):
+    """Weight function of the Laguerre polynomials.
+
+    The weight function is :math:`exp(-x)` and the interval of integration
+    is :math:`[0, \\inf]`. The Laguerre polynomials are orthogonal, but not
+    normalized, with respect to this weight function.
+
+    Parameters
+    ----------
+    x : array_like
+       Values at which the weight function will be computed.
+
+    Returns
+    -------
+    w : ndarray
+       The weight function at `x`.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagweight
+    >>> x = np.array([0, 1, 2])
+    >>> lagweight(x)
+    array([1.        , 0.36787944, 0.13533528])
+
+    """
+    w = np.exp(-x)
+    return w
+
+#
+# Laguerre series class
+#
+
+class Laguerre(ABCPolyBase):
+    """A Laguerre series class.
+
+    The Laguerre class provides the standard Python numerical methods
+    '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the
+    attributes and methods listed below.
+
+    Parameters
+    ----------
+    coef : array_like
+        Laguerre coefficients in order of increasing degree, i.e,
+        ``(1, 2, 3)`` gives ``1*L_0(x) + 2*L_1(X) + 3*L_2(x)``.
+    domain : (2,) array_like, optional
+        Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
+        to the interval ``[window[0], window[1]]`` by shifting and scaling.
+        The default value is [0., 1.].
+    window : (2,) array_like, optional
+        Window, see `domain` for its use. The default value is [0., 1.].
+    symbol : str, optional
+        Symbol used to represent the independent variable in string
+        representations of the polynomial expression, e.g. for printing.
+        The symbol must be a valid Python identifier. Default value is 'x'.
+
+        .. versionadded:: 1.24
+
+    """
+    # Virtual Functions
+    _add = staticmethod(lagadd)
+    _sub = staticmethod(lagsub)
+    _mul = staticmethod(lagmul)
+    _div = staticmethod(lagdiv)
+    _pow = staticmethod(lagpow)
+    _val = staticmethod(lagval)
+    _int = staticmethod(lagint)
+    _der = staticmethod(lagder)
+    _fit = staticmethod(lagfit)
+    _line = staticmethod(lagline)
+    _roots = staticmethod(lagroots)
+    _fromroots = staticmethod(lagfromroots)
+
+    # Virtual properties
+    domain = np.array(lagdomain)
+    window = np.array(lagdomain)
+    basis_name = 'L'

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy/polynomial/laguerre.pyi

@@ -0,0 +1,100 @@
+from typing import Any, ClassVar, Final, Literal as L
+
+import numpy as np
+
+from ._polybase import ABCPolyBase
+from ._polytypes import (
+    _Array1,
+    _Array2,
+    _FuncBinOp,
+    _FuncCompanion,
+    _FuncDer,
+    _FuncFit,
+    _FuncFromRoots,
+    _FuncGauss,
+    _FuncInteg,
+    _FuncLine,
+    _FuncPoly2Ortho,
+    _FuncPow,
+    _FuncRoots,
+    _FuncUnOp,
+    _FuncVal,
+    _FuncVal2D,
+    _FuncVal3D,
+    _FuncVander,
+    _FuncVander2D,
+    _FuncVander3D,
+    _FuncWeight,
+)
+from .polyutils import trimcoef as lagtrim
+
+__all__ = [
+    "lagzero",
+    "lagone",
+    "lagx",
+    "lagdomain",
+    "lagline",
+    "lagadd",
+    "lagsub",
+    "lagmulx",
+    "lagmul",
+    "lagdiv",
+    "lagpow",
+    "lagval",
+    "lagder",
+    "lagint",
+    "lag2poly",
+    "poly2lag",
+    "lagfromroots",
+    "lagvander",
+    "lagfit",
+    "lagtrim",
+    "lagroots",
+    "Laguerre",
+    "lagval2d",
+    "lagval3d",
+    "laggrid2d",
+    "laggrid3d",
+    "lagvander2d",
+    "lagvander3d",
+    "lagcompanion",
+    "laggauss",
+    "lagweight",
+]
+
+poly2lag: Final[_FuncPoly2Ortho] = ...
+lag2poly: Final[_FuncUnOp] = ...
+
+lagdomain: Final[_Array2[np.float64]] = ...
+lagzero: Final[_Array1[np.int_]] = ...
+lagone: Final[_Array1[np.int_]] = ...
+lagx: Final[_Array2[np.int_]] = ...
+
+lagline: Final[_FuncLine] = ...
+lagfromroots: Final[_FuncFromRoots] = ...
+lagadd: Final[_FuncBinOp] = ...
+lagsub: Final[_FuncBinOp] = ...
+lagmulx: Final[_FuncUnOp] = ...
+lagmul: Final[_FuncBinOp] = ...
+lagdiv: Final[_FuncBinOp] = ...
+lagpow: Final[_FuncPow] = ...
+lagder: Final[_FuncDer] = ...
+lagint: Final[_FuncInteg] = ...
+lagval: Final[_FuncVal] = ...
+lagval2d: Final[_FuncVal2D] = ...
+lagval3d: Final[_FuncVal3D] = ...
+laggrid2d: Final[_FuncVal2D] = ...
+laggrid3d: Final[_FuncVal3D] = ...
+lagvander: Final[_FuncVander] = ...
+lagvander2d: Final[_FuncVander2D] = ...
+lagvander3d: Final[_FuncVander3D] = ...
+lagfit: Final[_FuncFit] = ...
+lagcompanion: Final[_FuncCompanion] = ...
+lagroots: Final[_FuncRoots] = ...
+laggauss: Final[_FuncGauss] = ...
+lagweight: Final[_FuncWeight] = ...
+
+class Laguerre(ABCPolyBase[L["L"]]):
+    basis_name: ClassVar[L["L"]] = "L"  # pyright: ignore[reportIncompatibleMethodOverride]
+    domain: _Array2[np.float64 | Any] = ...  # pyright: ignore[reportIncompatibleMethodOverride]
+    window: _Array2[np.float64 | Any] = ...  # pyright: ignore[reportIncompatibleMethodOverride]

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy/polynomial/legendre.py

@@ -0,0 +1,1603 @@
+"""
+==================================================
+Legendre Series (:mod:`numpy.polynomial.legendre`)
+==================================================
+
+This module provides a number of objects (mostly functions) useful for
+dealing with Legendre series, including a `Legendre` class that
+encapsulates the usual arithmetic operations.  (General information
+on how this module represents and works with such polynomials is in the
+docstring for its "parent" sub-package, `numpy.polynomial`).
+
+Classes
+-------
+.. autosummary::
+   :toctree: generated/
+
+    Legendre
+
+Constants
+---------
+
+.. autosummary::
+   :toctree: generated/
+
+   legdomain
+   legzero
+   legone
+   legx
+
+Arithmetic
+----------
+
+.. autosummary::
+   :toctree: generated/
+
+   legadd
+   legsub
+   legmulx
+   legmul
+   legdiv
+   legpow
+   legval
+   legval2d
+   legval3d
+   leggrid2d
+   leggrid3d
+
+Calculus
+--------
+
+.. autosummary::
+   :toctree: generated/
+
+   legder
+   legint
+
+Misc Functions
+--------------
+
+.. autosummary::
+   :toctree: generated/
+
+   legfromroots
+   legroots
+   legvander
+   legvander2d
+   legvander3d
+   leggauss
+   legweight
+   legcompanion
+   legfit
+   legtrim
+   legline
+   leg2poly
+   poly2leg
+
+See also
+--------
+numpy.polynomial
+
+"""
+import numpy as np
+
+from . import polyutils as pu
+from ._polybase import ABCPolyBase
+
+__all__ = [
+    'legzero', 'legone', 'legx', 'legdomain', 'legline', 'legadd',
+    'legsub', 'legmulx', 'legmul', 'legdiv', 'legpow', 'legval', 'legder',
+    'legint', 'leg2poly', 'poly2leg', 'legfromroots', 'legvander',
+    'legfit', 'legtrim', 'legroots', 'Legendre', 'legval2d', 'legval3d',
+    'leggrid2d', 'leggrid3d', 'legvander2d', 'legvander3d', 'legcompanion',
+    'leggauss', 'legweight']
+
+legtrim = pu.trimcoef
+
+
+def poly2leg(pol):
+    """
+    Convert a polynomial to a Legendre series.
+
+    Convert an array representing the coefficients of a polynomial (relative
+    to the "standard" basis) ordered from lowest degree to highest, to an
+    array of the coefficients of the equivalent Legendre series, ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    pol : array_like
+        1-D array containing the polynomial coefficients
+
+    Returns
+    -------
+    c : ndarray
+        1-D array containing the coefficients of the equivalent Legendre
+        series.
+
+    See Also
+    --------
+    leg2poly
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from numpy import polynomial as P
+    >>> p = P.Polynomial(np.arange(4))
+    >>> p
+    Polynomial([0.,  1.,  2.,  3.], domain=[-1.,  1.], window=[-1.,  1.], ...
+    >>> c = P.Legendre(P.legendre.poly2leg(p.coef))
+    >>> c
+    Legendre([ 1.  ,  3.25,  1.  ,  0.75], domain=[-1,  1], window=[-1,  1]) # may vary
+
+    """
+    [pol] = pu.as_series([pol])
+    deg = len(pol) - 1
+    res = 0
+    for i in range(deg, -1, -1):
+        res = legadd(legmulx(res), pol[i])
+    return res
+
+
+def leg2poly(c):
+    """
+    Convert a Legendre series to a polynomial.
+
+    Convert an array representing the coefficients of a Legendre series,
+    ordered from lowest degree to highest, to an array of the coefficients
+    of the equivalent polynomial (relative to the "standard" basis) ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array containing the Legendre series coefficients, ordered
+        from lowest order term to highest.
+
+    Returns
+    -------
+    pol : ndarray
+        1-D array containing the coefficients of the equivalent polynomial
+        (relative to the "standard" basis) ordered from lowest order term
+        to highest.
+
+    See Also
+    --------
+    poly2leg
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> from numpy import polynomial as P
+    >>> c = P.Legendre(range(4))
+    >>> c
+    Legendre([0., 1., 2., 3.], domain=[-1.,  1.], window=[-1.,  1.], symbol='x')
+    >>> p = c.convert(kind=P.Polynomial)
+    >>> p
+    Polynomial([-1. , -3.5,  3. ,  7.5], domain=[-1.,  1.], window=[-1., ...
+    >>> P.legendre.leg2poly(range(4))
+    array([-1. , -3.5,  3. ,  7.5])
+
+
+    """
+    from .polynomial import polyadd, polymulx, polysub
+
+    [c] = pu.as_series([c])
+    n = len(c)
+    if n < 3:
+        return c
+    else:
+        c0 = c[-2]
+        c1 = c[-1]
+        # i is the current degree of c1
+        for i in range(n - 1, 1, -1):
+            tmp = c0
+            c0 = polysub(c[i - 2], (c1 * (i - 1)) / i)
+            c1 = polyadd(tmp, (polymulx(c1) * (2 * i - 1)) / i)
+        return polyadd(c0, polymulx(c1))
+
+
+#
+# These are constant arrays are of integer type so as to be compatible
+# with the widest range of other types, such as Decimal.
+#
+
+# Legendre
+legdomain = np.array([-1., 1.])
+
+# Legendre coefficients representing zero.
+legzero = np.array([0])
+
+# Legendre coefficients representing one.
+legone = np.array([1])
+
+# Legendre coefficients representing the identity x.
+legx = np.array([0, 1])
+
+
+def legline(off, scl):
+    """
+    Legendre series whose graph is a straight line.
+
+
+
+    Parameters
+    ----------
+    off, scl : scalars
+        The specified line is given by ``off + scl*x``.
+
+    Returns
+    -------
+    y : ndarray
+        This module's representation of the Legendre series for
+        ``off + scl*x``.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyline
+    numpy.polynomial.chebyshev.chebline
+    numpy.polynomial.laguerre.lagline
+    numpy.polynomial.hermite.hermline
+    numpy.polynomial.hermite_e.hermeline
+
+    Examples
+    --------
+    >>> import numpy.polynomial.legendre as L
+    >>> L.legline(3,2)
+    array([3, 2])
+    >>> L.legval(-3, L.legline(3,2)) # should be -3
+    -3.0
+
+    """
+    if scl != 0:
+        return np.array([off, scl])
+    else:
+        return np.array([off])
+
+
+def legfromroots(roots):
+    """
+    Generate a Legendre series with given roots.
+
+    The function returns the coefficients of the polynomial
+
+    .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
+
+    in Legendre form, where the :math:`r_n` are the roots specified in `roots`.
+    If a zero has multiplicity n, then it must appear in `roots` n times.
+    For instance, if 2 is a root of multiplicity three and 3 is a root of
+    multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
+    roots can appear in any order.
+
+    If the returned coefficients are `c`, then
+
+    .. math:: p(x) = c_0 + c_1 * L_1(x) + ... +  c_n * L_n(x)
+
+    The coefficient of the last term is not generally 1 for monic
+    polynomials in Legendre form.
+
+    Parameters
+    ----------
+    roots : array_like
+        Sequence containing the roots.
+
+    Returns
+    -------
+    out : ndarray
+        1-D array of coefficients.  If all roots are real then `out` is a
+        real array, if some of the roots are complex, then `out` is complex
+        even if all the coefficients in the result are real (see Examples
+        below).
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyfromroots
+    numpy.polynomial.chebyshev.chebfromroots
+    numpy.polynomial.laguerre.lagfromroots
+    numpy.polynomial.hermite.hermfromroots
+    numpy.polynomial.hermite_e.hermefromroots
+
+    Examples
+    --------
+    >>> import numpy.polynomial.legendre as L
+    >>> L.legfromroots((-1,0,1)) # x^3 - x relative to the standard basis
+    array([ 0. , -0.4,  0. ,  0.4])
+    >>> j = complex(0,1)
+    >>> L.legfromroots((-j,j)) # x^2 + 1 relative to the standard basis
+    array([ 1.33333333+0.j,  0.00000000+0.j,  0.66666667+0.j]) # may vary
+
+    """
+    return pu._fromroots(legline, legmul, roots)
+
+
+def legadd(c1, c2):
+    """
+    Add one Legendre series to another.
+
+    Returns the sum of two Legendre series `c1` + `c2`.  The arguments
+    are sequences of coefficients ordered from lowest order term to
+    highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Legendre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the Legendre series of their sum.
+
+    See Also
+    --------
+    legsub, legmulx, legmul, legdiv, legpow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the sum of two Legendre series
+    is a Legendre series (without having to "reproject" the result onto
+    the basis set) so addition, just like that of "standard" polynomials,
+    is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial import legendre as L
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2,1)
+    >>> L.legadd(c1,c2)
+    array([4.,  4.,  4.])
+
+    """
+    return pu._add(c1, c2)
+
+
+def legsub(c1, c2):
+    """
+    Subtract one Legendre series from another.
+
+    Returns the difference of two Legendre series `c1` - `c2`.  The
+    sequences of coefficients are from lowest order term to highest, i.e.,
+    [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Legendre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Legendre series coefficients representing their difference.
+
+    See Also
+    --------
+    legadd, legmulx, legmul, legdiv, legpow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the difference of two Legendre
+    series is a Legendre series (without having to "reproject" the result
+    onto the basis set) so subtraction, just like that of "standard"
+    polynomials, is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial import legendre as L
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2,1)
+    >>> L.legsub(c1,c2)
+    array([-2.,  0.,  2.])
+    >>> L.legsub(c2,c1) # -C.legsub(c1,c2)
+    array([ 2.,  0., -2.])
+
+    """
+    return pu._sub(c1, c2)
+
+
+def legmulx(c):
+    """Multiply a Legendre series by x.
+
+    Multiply the Legendre series `c` by x, where x is the independent
+    variable.
+
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Legendre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the result of the multiplication.
+
+    See Also
+    --------
+    legadd, legsub, legmul, legdiv, legpow
+
+    Notes
+    -----
+    The multiplication uses the recursion relationship for Legendre
+    polynomials in the form
+
+    .. math::
+
+      xP_i(x) = ((i + 1)*P_{i + 1}(x) + i*P_{i - 1}(x))/(2i + 1)
+
+    Examples
+    --------
+    >>> from numpy.polynomial import legendre as L
+    >>> L.legmulx([1,2,3])
+    array([ 0.66666667, 2.2, 1.33333333, 1.8]) # may vary
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    # The zero series needs special treatment
+    if len(c) == 1 and c[0] == 0:
+        return c
+
+    prd = np.empty(len(c) + 1, dtype=c.dtype)
+    prd[0] = c[0] * 0
+    prd[1] = c[0]
+    for i in range(1, len(c)):
+        j = i + 1
+        k = i - 1
+        s = i + j
+        prd[j] = (c[i] * j) / s
+        prd[k] += (c[i] * i) / s
+    return prd
+
+
+def legmul(c1, c2):
+    """
+    Multiply one Legendre series by another.
+
+    Returns the product of two Legendre series `c1` * `c2`.  The arguments
+    are sequences of coefficients, from lowest order "term" to highest,
+    e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Legendre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Legendre series coefficients representing their product.
+
+    See Also
+    --------
+    legadd, legsub, legmulx, legdiv, legpow
+
+    Notes
+    -----
+    In general, the (polynomial) product of two C-series results in terms
+    that are not in the Legendre polynomial basis set.  Thus, to express
+    the product as a Legendre series, it is necessary to "reproject" the
+    product onto said basis set, which may produce "unintuitive" (but
+    correct) results; see Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import legendre as L
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2)
+    >>> L.legmul(c1,c2) # multiplication requires "reprojection"
+    array([  4.33333333,  10.4       ,  11.66666667,   3.6       ]) # may vary
+
+    """
+    # s1, s2 are trimmed copies
+    [c1, c2] = pu.as_series([c1, c2])
+
+    if len(c1) > len(c2):
+        c = c2
+        xs = c1
+    else:
+        c = c1
+        xs = c2
+
+    if len(c) == 1:
+        c0 = c[0] * xs
+        c1 = 0
+    elif len(c) == 2:
+        c0 = c[0] * xs
+        c1 = c[1] * xs
+    else:
+        nd = len(c)
+        c0 = c[-2] * xs
+        c1 = c[-1] * xs
+        for i in range(3, len(c) + 1):
+            tmp = c0
+            nd = nd - 1
+            c0 = legsub(c[-i] * xs, (c1 * (nd - 1)) / nd)
+            c1 = legadd(tmp, (legmulx(c1) * (2 * nd - 1)) / nd)
+    return legadd(c0, legmulx(c1))
+
+
+def legdiv(c1, c2):
+    """
+    Divide one Legendre series by another.
+
+    Returns the quotient-with-remainder of two Legendre series
+    `c1` / `c2`.  The arguments are sequences of coefficients from lowest
+    order "term" to highest, e.g., [1,2,3] represents the series
+    ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Legendre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    quo, rem : ndarrays
+        Of Legendre series coefficients representing the quotient and
+        remainder.
+
+    See Also
+    --------
+    legadd, legsub, legmulx, legmul, legpow
+
+    Notes
+    -----
+    In general, the (polynomial) division of one Legendre series by another
+    results in quotient and remainder terms that are not in the Legendre
+    polynomial basis set.  Thus, to express these results as a Legendre
+    series, it is necessary to "reproject" the results onto the Legendre
+    basis set, which may produce "unintuitive" (but correct) results; see
+    Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import legendre as L
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2,1)
+    >>> L.legdiv(c1,c2) # quotient "intuitive," remainder not
+    (array([3.]), array([-8., -4.]))
+    >>> c2 = (0,1,2,3)
+    >>> L.legdiv(c2,c1) # neither "intuitive"
+    (array([-0.07407407,  1.66666667]), array([-1.03703704, -2.51851852])) # may vary
+
+    """
+    return pu._div(legmul, c1, c2)
+
+
+def legpow(c, pow, maxpower=16):
+    """Raise a Legendre series to a power.
+
+    Returns the Legendre series `c` raised to the power `pow`. The
+    argument `c` is a sequence of coefficients ordered from low to high.
+    i.e., [1,2,3] is the series  ``P_0 + 2*P_1 + 3*P_2.``
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Legendre series coefficients ordered from low to
+        high.
+    pow : integer
+        Power to which the series will be raised
+    maxpower : integer, optional
+        Maximum power allowed. This is mainly to limit growth of the series
+        to unmanageable size. Default is 16
+
+    Returns
+    -------
+    coef : ndarray
+        Legendre series of power.
+
+    See Also
+    --------
+    legadd, legsub, legmulx, legmul, legdiv
+
+    """
+    return pu._pow(legmul, c, pow, maxpower)
+
+
+def legder(c, m=1, scl=1, axis=0):
+    """
+    Differentiate a Legendre series.
+
+    Returns the Legendre series coefficients `c` differentiated `m` times
+    along `axis`.  At each iteration the result is multiplied by `scl` (the
+    scaling factor is for use in a linear change of variable). The argument
+    `c` is an array of coefficients from low to high degree along each
+    axis, e.g., [1,2,3] represents the series ``1*L_0 + 2*L_1 + 3*L_2``
+    while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) +
+    2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is
+    ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Legendre series coefficients. If c is multidimensional the
+        different axis correspond to different variables with the degree in
+        each axis given by the corresponding index.
+    m : int, optional
+        Number of derivatives taken, must be non-negative. (Default: 1)
+    scl : scalar, optional
+        Each differentiation is multiplied by `scl`.  The end result is
+        multiplication by ``scl**m``.  This is for use in a linear change of
+        variable. (Default: 1)
+    axis : int, optional
+        Axis over which the derivative is taken. (Default: 0).
+
+    Returns
+    -------
+    der : ndarray
+        Legendre series of the derivative.
+
+    See Also
+    --------
+    legint
+
+    Notes
+    -----
+    In general, the result of differentiating a Legendre series does not
+    resemble the same operation on a power series. Thus the result of this
+    function may be "unintuitive," albeit correct; see Examples section
+    below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import legendre as L
+    >>> c = (1,2,3,4)
+    >>> L.legder(c)
+    array([  6.,   9.,  20.])
+    >>> L.legder(c, 3)
+    array([60.])
+    >>> L.legder(c, scl=-1)
+    array([ -6.,  -9., -20.])
+    >>> L.legder(c, 2,-1)
+    array([  9.,  60.])
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    cnt = pu._as_int(m, "the order of derivation")
+    iaxis = pu._as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of derivation must be non-negative")
+    iaxis = np.lib.array_utils.normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    n = len(c)
+    if cnt >= n:
+        c = c[:1] * 0
+    else:
+        for i in range(cnt):
+            n = n - 1
+            c *= scl
+            der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
+            for j in range(n, 2, -1):
+                der[j - 1] = (2 * j - 1) * c[j]
+                c[j - 2] += c[j]
+            if n > 1:
+                der[1] = 3 * c[2]
+            der[0] = c[1]
+            c = der
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def legint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
+    """
+    Integrate a Legendre series.
+
+    Returns the Legendre series coefficients `c` integrated `m` times from
+    `lbnd` along `axis`. At each iteration the resulting series is
+    **multiplied** by `scl` and an integration constant, `k`, is added.
+    The scaling factor is for use in a linear change of variable.  ("Buyer
+    beware": note that, depending on what one is doing, one may want `scl`
+    to be the reciprocal of what one might expect; for more information,
+    see the Notes section below.)  The argument `c` is an array of
+    coefficients from low to high degree along each axis, e.g., [1,2,3]
+    represents the series ``L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]]
+    represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) +
+    2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Legendre series coefficients. If c is multidimensional the
+        different axis correspond to different variables with the degree in
+        each axis given by the corresponding index.
+    m : int, optional
+        Order of integration, must be positive. (Default: 1)
+    k : {[], list, scalar}, optional
+        Integration constant(s).  The value of the first integral at
+        ``lbnd`` is the first value in the list, the value of the second
+        integral at ``lbnd`` is the second value, etc.  If ``k == []`` (the
+        default), all constants are set to zero.  If ``m == 1``, a single
+        scalar can be given instead of a list.
+    lbnd : scalar, optional
+        The lower bound of the integral. (Default: 0)
+    scl : scalar, optional
+        Following each integration the result is *multiplied* by `scl`
+        before the integration constant is added. (Default: 1)
+    axis : int, optional
+        Axis over which the integral is taken. (Default: 0).
+
+    Returns
+    -------
+    S : ndarray
+        Legendre series coefficient array of the integral.
+
+    Raises
+    ------
+    ValueError
+        If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
+        ``np.ndim(scl) != 0``.
+
+    See Also
+    --------
+    legder
+
+    Notes
+    -----
+    Note that the result of each integration is *multiplied* by `scl`.
+    Why is this important to note?  Say one is making a linear change of
+    variable :math:`u = ax + b` in an integral relative to `x`.  Then
+    :math:`dx = du/a`, so one will need to set `scl` equal to
+    :math:`1/a` - perhaps not what one would have first thought.
+
+    Also note that, in general, the result of integrating a C-series needs
+    to be "reprojected" onto the C-series basis set.  Thus, typically,
+    the result of this function is "unintuitive," albeit correct; see
+    Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import legendre as L
+    >>> c = (1,2,3)
+    >>> L.legint(c)
+    array([ 0.33333333,  0.4       ,  0.66666667,  0.6       ]) # may vary
+    >>> L.legint(c, 3)
+    array([  1.66666667e-02,  -1.78571429e-02,   4.76190476e-02, # may vary
+             -1.73472348e-18,   1.90476190e-02,   9.52380952e-03])
+    >>> L.legint(c, k=3)
+     array([ 3.33333333,  0.4       ,  0.66666667,  0.6       ]) # may vary
+    >>> L.legint(c, lbnd=-2)
+    array([ 7.33333333,  0.4       ,  0.66666667,  0.6       ]) # may vary
+    >>> L.legint(c, scl=2)
+    array([ 0.66666667,  0.8       ,  1.33333333,  1.2       ]) # may vary
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if not np.iterable(k):
+        k = [k]
+    cnt = pu._as_int(m, "the order of integration")
+    iaxis = pu._as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of integration must be non-negative")
+    if len(k) > cnt:
+        raise ValueError("Too many integration constants")
+    if np.ndim(lbnd) != 0:
+        raise ValueError("lbnd must be a scalar.")
+    if np.ndim(scl) != 0:
+        raise ValueError("scl must be a scalar.")
+    iaxis = np.lib.array_utils.normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    k = list(k) + [0] * (cnt - len(k))
+    for i in range(cnt):
+        n = len(c)
+        c *= scl
+        if n == 1 and np.all(c[0] == 0):
+            c[0] += k[i]
+        else:
+            tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype)
+            tmp[0] = c[0] * 0
+            tmp[1] = c[0]
+            if n > 1:
+                tmp[2] = c[1] / 3
+            for j in range(2, n):
+                t = c[j] / (2 * j + 1)
+                tmp[j + 1] = t
+                tmp[j - 1] -= t
+            tmp[0] += k[i] - legval(lbnd, tmp)
+            c = tmp
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def legval(x, c, tensor=True):
+    """
+    Evaluate a Legendre series at points x.
+
+    If `c` is of length ``n + 1``, this function returns the value:
+
+    .. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)
+
+    The parameter `x` is converted to an array only if it is a tuple or a
+    list, otherwise it is treated as a scalar. In either case, either `x`
+    or its elements must support multiplication and addition both with
+    themselves and with the elements of `c`.
+
+    If `c` is a 1-D array, then ``p(x)`` will have the same shape as `x`.  If
+    `c` is multidimensional, then the shape of the result depends on the
+    value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
+    x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
+    scalars have shape (,).
+
+    Trailing zeros in the coefficients will be used in the evaluation, so
+    they should be avoided if efficiency is a concern.
+
+    Parameters
+    ----------
+    x : array_like, compatible object
+        If `x` is a list or tuple, it is converted to an ndarray, otherwise
+        it is left unchanged and treated as a scalar. In either case, `x`
+        or its elements must support addition and multiplication with
+        themselves and with the elements of `c`.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree n are contained in c[n]. If `c` is multidimensional the
+        remaining indices enumerate multiple polynomials. In the two
+        dimensional case the coefficients may be thought of as stored in
+        the columns of `c`.
+    tensor : boolean, optional
+        If True, the shape of the coefficient array is extended with ones
+        on the right, one for each dimension of `x`. Scalars have dimension 0
+        for this action. The result is that every column of coefficients in
+        `c` is evaluated for every element of `x`. If False, `x` is broadcast
+        over the columns of `c` for the evaluation.  This keyword is useful
+        when `c` is multidimensional. The default value is True.
+
+    Returns
+    -------
+    values : ndarray, algebra_like
+        The shape of the return value is described above.
+
+    See Also
+    --------
+    legval2d, leggrid2d, legval3d, leggrid3d
+
+    Notes
+    -----
+    The evaluation uses Clenshaw recursion, aka synthetic division.
+
+    """
+    c = np.array(c, ndmin=1, copy=None)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if isinstance(x, (tuple, list)):
+        x = np.asarray(x)
+    if isinstance(x, np.ndarray) and tensor:
+        c = c.reshape(c.shape + (1,) * x.ndim)
+
+    if len(c) == 1:
+        c0 = c[0]
+        c1 = 0
+    elif len(c) == 2:
+        c0 = c[0]
+        c1 = c[1]
+    else:
+        nd = len(c)
+        c0 = c[-2]
+        c1 = c[-1]
+        for i in range(3, len(c) + 1):
+            tmp = c0
+            nd = nd - 1
+            c0 = c[-i] - c1 * ((nd - 1) / nd)
+            c1 = tmp + c1 * x * ((2 * nd - 1) / nd)
+    return c0 + c1 * x
+
+
+def legval2d(x, y, c):
+    """
+    Evaluate a 2-D Legendre series at points (x, y).
+
+    This function returns the values:
+
+    .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_j(y)
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars and they
+    must have the same shape after conversion. In either case, either `x`
+    and `y` or their elements must support multiplication and addition both
+    with themselves and with the elements of `c`.
+
+    If `c` is a 1-D array a one is implicitly appended to its shape to make
+    it 2-D. The shape of the result will be c.shape[2:] + x.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points ``(x, y)``,
+        where `x` and `y` must have the same shape. If `x` or `y` is a list
+        or tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and if it isn't an ndarray it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term
+        of multi-degree i,j is contained in ``c[i,j]``. If `c` has
+        dimension greater than two the remaining indices enumerate multiple
+        sets of coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional Legendre series at points formed
+        from pairs of corresponding values from `x` and `y`.
+
+    See Also
+    --------
+    legval, leggrid2d, legval3d, leggrid3d
+    """
+    return pu._valnd(legval, c, x, y)
+
+
+def leggrid2d(x, y, c):
+    """
+    Evaluate a 2-D Legendre series on the Cartesian product of x and y.
+
+    This function returns the values:
+
+    .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * L_i(a) * L_j(b)
+
+    where the points ``(a, b)`` consist of all pairs formed by taking
+    `a` from `x` and `b` from `y`. The resulting points form a grid with
+    `x` in the first dimension and `y` in the second.
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars. In either
+    case, either `x` and `y` or their elements must support multiplication
+    and addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than two dimensions, ones are implicitly appended to
+    its shape to make it 2-D. The shape of the result will be c.shape[2:] +
+    x.shape + y.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points in the
+        Cartesian product of `x` and `y`.  If `x` or `y` is a list or
+        tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and, if it isn't an ndarray, it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term of
+        multi-degree i,j is contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional Chebyshev series at points in the
+        Cartesian product of `x` and `y`.
+
+    See Also
+    --------
+    legval, legval2d, legval3d, leggrid3d
+    """
+    return pu._gridnd(legval, c, x, y)
+
+
+def legval3d(x, y, z, c):
+    """
+    Evaluate a 3-D Legendre series at points (x, y, z).
+
+    This function returns the values:
+
+    .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_k(z)
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if
+    they are tuples or a lists, otherwise they are treated as a scalars and
+    they must have the same shape after conversion. In either case, either
+    `x`, `y`, and `z` or their elements must support multiplication and
+    addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than 3 dimensions, ones are implicitly appended to its
+    shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible object
+        The three dimensional series is evaluated at the points
+        ``(x, y, z)``, where `x`, `y`, and `z` must have the same shape.  If
+        any of `x`, `y`, or `z` is a list or tuple, it is first converted
+        to an ndarray, otherwise it is left unchanged and if it isn't an
+        ndarray it is  treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term of
+        multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
+        greater than 3 the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the multidimensional polynomial on points formed with
+        triples of corresponding values from `x`, `y`, and `z`.
+
+    See Also
+    --------
+    legval, legval2d, leggrid2d, leggrid3d
+    """
+    return pu._valnd(legval, c, x, y, z)
+
+
+def leggrid3d(x, y, z, c):
+    """
+    Evaluate a 3-D Legendre series on the Cartesian product of x, y, and z.
+
+    This function returns the values:
+
+    .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * L_i(a) * L_j(b) * L_k(c)
+
+    where the points ``(a, b, c)`` consist of all triples formed by taking
+    `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
+    a grid with `x` in the first dimension, `y` in the second, and `z` in
+    the third.
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if they
+    are tuples or a lists, otherwise they are treated as a scalars. In
+    either case, either `x`, `y`, and `z` or their elements must support
+    multiplication and addition both with themselves and with the elements
+    of `c`.
+
+    If `c` has fewer than three dimensions, ones are implicitly appended to
+    its shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape + y.shape + z.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible objects
+        The three dimensional series is evaluated at the points in the
+        Cartesian product of `x`, `y`, and `z`.  If `x`, `y`, or `z` is a
+        list or tuple, it is first converted to an ndarray, otherwise it is
+        left unchanged and, if it isn't an ndarray, it is treated as a
+        scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree i,j are contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points in the Cartesian
+        product of `x` and `y`.
+
+    See Also
+    --------
+    legval, legval2d, leggrid2d, legval3d
+    """
+    return pu._gridnd(legval, c, x, y, z)
+
+
+def legvander(x, deg):
+    """Pseudo-Vandermonde matrix of given degree.
+
+    Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
+    `x`. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., i] = L_i(x)
+
+    where ``0 <= i <= deg``. The leading indices of `V` index the elements of
+    `x` and the last index is the degree of the Legendre polynomial.
+
+    If `c` is a 1-D array of coefficients of length ``n + 1`` and `V` is the
+    array ``V = legvander(x, n)``, then ``np.dot(V, c)`` and
+    ``legval(x, c)`` are the same up to roundoff. This equivalence is
+    useful both for least squares fitting and for the evaluation of a large
+    number of Legendre series of the same degree and sample points.
+
+    Parameters
+    ----------
+    x : array_like
+        Array of points. The dtype is converted to float64 or complex128
+        depending on whether any of the elements are complex. If `x` is
+        scalar it is converted to a 1-D array.
+    deg : int
+        Degree of the resulting matrix.
+
+    Returns
+    -------
+    vander : ndarray
+        The pseudo-Vandermonde matrix. The shape of the returned matrix is
+        ``x.shape + (deg + 1,)``, where The last index is the degree of the
+        corresponding Legendre polynomial.  The dtype will be the same as
+        the converted `x`.
+
+    """
+    ideg = pu._as_int(deg, "deg")
+    if ideg < 0:
+        raise ValueError("deg must be non-negative")
+
+    x = np.array(x, copy=None, ndmin=1) + 0.0
+    dims = (ideg + 1,) + x.shape
+    dtyp = x.dtype
+    v = np.empty(dims, dtype=dtyp)
+    # Use forward recursion to generate the entries. This is not as accurate
+    # as reverse recursion in this application but it is more efficient.
+    v[0] = x * 0 + 1
+    if ideg > 0:
+        v[1] = x
+        for i in range(2, ideg + 1):
+            v[i] = (v[i - 1] * x * (2 * i - 1) - v[i - 2] * (i - 1)) / i
+    return np.moveaxis(v, 0, -1)
+
+
+def legvander2d(x, y, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points ``(x, y)``. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (deg[1] + 1)*i + j] = L_i(x) * L_j(y),
+
+    where ``0 <= i <= deg[0]`` and ``0 <= j <= deg[1]``. The leading indices of
+    `V` index the points ``(x, y)`` and the last index encodes the degrees of
+    the Legendre polynomials.
+
+    If ``V = legvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
+    correspond to the elements of a 2-D coefficient array `c` of shape
+    (xdeg + 1, ydeg + 1) in the order
+
+    .. math:: c_{00}, c_{01}, c_{02}, ... , c_{10}, c_{11}, c_{12}, ...
+
+    and ``np.dot(V, c.flat)`` and ``legval2d(x, y, c)`` will be the same
+    up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 2-D Legendre
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes
+        will be converted to either float64 or complex128 depending on
+        whether any of the elements are complex. Scalars are converted to
+        1-D arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg].
+
+    Returns
+    -------
+    vander2d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)`.  The dtype will be the same
+        as the converted `x` and `y`.
+
+    See Also
+    --------
+    legvander, legvander3d, legval2d, legval3d
+    """
+    return pu._vander_nd_flat((legvander, legvander), (x, y), deg)
+
+
+def legvander3d(x, y, z, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points ``(x, y, z)``. If `l`, `m`, `n` are the given degrees in `x`, `y`, `z`,
+    then The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = L_i(x)*L_j(y)*L_k(z),
+
+    where ``0 <= i <= l``, ``0 <= j <= m``, and ``0 <= j <= n``.  The leading
+    indices of `V` index the points ``(x, y, z)`` and the last index encodes
+    the degrees of the Legendre polynomials.
+
+    If ``V = legvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
+    of `V` correspond to the elements of a 3-D coefficient array `c` of
+    shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
+
+    .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
+
+    and ``np.dot(V, c.flat)`` and ``legval3d(x, y, z, c)`` will be the
+    same up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 3-D Legendre
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y, z : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes will
+        be converted to either float64 or complex128 depending on whether
+        any of the elements are complex. Scalars are converted to 1-D
+        arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg, z_deg].
+
+    Returns
+    -------
+    vander3d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`.  The dtype will
+        be the same as the converted `x`, `y`, and `z`.
+
+    See Also
+    --------
+    legvander, legvander3d, legval2d, legval3d
+    """
+    return pu._vander_nd_flat((legvander, legvander, legvander), (x, y, z), deg)
+
+
+def legfit(x, y, deg, rcond=None, full=False, w=None):
+    """
+    Least squares fit of Legendre series to data.
+
+    Return the coefficients of a Legendre series of degree `deg` that is the
+    least squares fit to the data values `y` given at points `x`. If `y` is
+    1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
+    fits are done, one for each column of `y`, and the resulting
+    coefficients are stored in the corresponding columns of a 2-D return.
+    The fitted polynomial(s) are in the form
+
+    .. math::  p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x),
+
+    where `n` is `deg`.
+
+    Parameters
+    ----------
+    x : array_like, shape (M,)
+        x-coordinates of the M sample points ``(x[i], y[i])``.
+    y : array_like, shape (M,) or (M, K)
+        y-coordinates of the sample points. Several data sets of sample
+        points sharing the same x-coordinates can be fitted at once by
+        passing in a 2D-array that contains one dataset per column.
+    deg : int or 1-D array_like
+        Degree(s) of the fitting polynomials. If `deg` is a single integer
+        all terms up to and including the `deg`'th term are included in the
+        fit. For NumPy versions >= 1.11.0 a list of integers specifying the
+        degrees of the terms to include may be used instead.
+    rcond : float, optional
+        Relative condition number of the fit. Singular values smaller than
+        this relative to the largest singular value will be ignored. The
+        default value is len(x)*eps, where eps is the relative precision of
+        the float type, about 2e-16 in most cases.
+    full : bool, optional
+        Switch determining nature of return value. When it is False (the
+        default) just the coefficients are returned, when True diagnostic
+        information from the singular value decomposition is also returned.
+    w : array_like, shape (`M`,), optional
+        Weights. If not None, the weight ``w[i]`` applies to the unsquared
+        residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
+        chosen so that the errors of the products ``w[i]*y[i]`` all have the
+        same variance.  When using inverse-variance weighting, use
+        ``w[i] = 1/sigma(y[i])``.  The default value is None.
+
+    Returns
+    -------
+    coef : ndarray, shape (M,) or (M, K)
+        Legendre coefficients ordered from low to high. If `y` was
+        2-D, the coefficients for the data in column k of `y` are in
+        column `k`. If `deg` is specified as a list, coefficients for
+        terms not included in the fit are set equal to zero in the
+        returned `coef`.
+
+    [residuals, rank, singular_values, rcond] : list
+        These values are only returned if ``full == True``
+
+        - residuals -- sum of squared residuals of the least squares fit
+        - rank -- the numerical rank of the scaled Vandermonde matrix
+        - singular_values -- singular values of the scaled Vandermonde matrix
+        - rcond -- value of `rcond`.
+
+        For more details, see `numpy.linalg.lstsq`.
+
+    Warns
+    -----
+    RankWarning
+        The rank of the coefficient matrix in the least-squares fit is
+        deficient. The warning is only raised if ``full == False``.  The
+        warnings can be turned off by
+
+        >>> import warnings
+        >>> warnings.simplefilter('ignore', np.exceptions.RankWarning)
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyfit
+    numpy.polynomial.chebyshev.chebfit
+    numpy.polynomial.laguerre.lagfit
+    numpy.polynomial.hermite.hermfit
+    numpy.polynomial.hermite_e.hermefit
+    legval : Evaluates a Legendre series.
+    legvander : Vandermonde matrix of Legendre series.
+    legweight : Legendre weight function (= 1).
+    numpy.linalg.lstsq : Computes a least-squares fit from the matrix.
+    scipy.interpolate.UnivariateSpline : Computes spline fits.
+
+    Notes
+    -----
+    The solution is the coefficients of the Legendre series `p` that
+    minimizes the sum of the weighted squared errors
+
+    .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
+
+    where :math:`w_j` are the weights. This problem is solved by setting up
+    as the (typically) overdetermined matrix equation
+
+    .. math:: V(x) * c = w * y,
+
+    where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the
+    coefficients to be solved for, `w` are the weights, and `y` are the
+    observed values.  This equation is then solved using the singular value
+    decomposition of `V`.
+
+    If some of the singular values of `V` are so small that they are
+    neglected, then a `~exceptions.RankWarning` will be issued. This means that
+    the coefficient values may be poorly determined. Using a lower order fit
+    will usually get rid of the warning.  The `rcond` parameter can also be
+    set to a value smaller than its default, but the resulting fit may be
+    spurious and have large contributions from roundoff error.
+
+    Fits using Legendre series are usually better conditioned than fits
+    using power series, but much can depend on the distribution of the
+    sample points and the smoothness of the data. If the quality of the fit
+    is inadequate splines may be a good alternative.
+
+    References
+    ----------
+    .. [1] Wikipedia, "Curve fitting",
+           https://en.wikipedia.org/wiki/Curve_fitting
+
+    Examples
+    --------
+
+    """
+    return pu._fit(legvander, x, y, deg, rcond, full, w)
+
+
+def legcompanion(c):
+    """Return the scaled companion matrix of c.
+
+    The basis polynomials are scaled so that the companion matrix is
+    symmetric when `c` is an Legendre basis polynomial. This provides
+    better eigenvalue estimates than the unscaled case and for basis
+    polynomials the eigenvalues are guaranteed to be real if
+    `numpy.linalg.eigvalsh` is used to obtain them.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Legendre series coefficients ordered from low to high
+        degree.
+
+    Returns
+    -------
+    mat : ndarray
+        Scaled companion matrix of dimensions (deg, deg).
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) < 2:
+        raise ValueError('Series must have maximum degree of at least 1.')
+    if len(c) == 2:
+        return np.array([[-c[0] / c[1]]])
+
+    n = len(c) - 1
+    mat = np.zeros((n, n), dtype=c.dtype)
+    scl = 1. / np.sqrt(2 * np.arange(n) + 1)
+    top = mat.reshape(-1)[1::n + 1]
+    bot = mat.reshape(-1)[n::n + 1]
+    top[...] = np.arange(1, n) * scl[:n - 1] * scl[1:n]
+    bot[...] = top
+    mat[:, -1] -= (c[:-1] / c[-1]) * (scl / scl[-1]) * (n / (2 * n - 1))
+    return mat
+
+
+def legroots(c):
+    """
+    Compute the roots of a Legendre series.
+
+    Return the roots (a.k.a. "zeros") of the polynomial
+
+    .. math:: p(x) = \\sum_i c[i] * L_i(x).
+
+    Parameters
+    ----------
+    c : 1-D array_like
+        1-D array of coefficients.
+
+    Returns
+    -------
+    out : ndarray
+        Array of the roots of the series. If all the roots are real,
+        then `out` is also real, otherwise it is complex.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyroots
+    numpy.polynomial.chebyshev.chebroots
+    numpy.polynomial.laguerre.lagroots
+    numpy.polynomial.hermite.hermroots
+    numpy.polynomial.hermite_e.hermeroots
+
+    Notes
+    -----
+    The root estimates are obtained as the eigenvalues of the companion
+    matrix, Roots far from the origin of the complex plane may have large
+    errors due to the numerical instability of the series for such values.
+    Roots with multiplicity greater than 1 will also show larger errors as
+    the value of the series near such points is relatively insensitive to
+    errors in the roots. Isolated roots near the origin can be improved by
+    a few iterations of Newton's method.
+
+    The Legendre series basis polynomials aren't powers of ``x`` so the
+    results of this function may seem unintuitive.
+
+    Examples
+    --------
+    >>> import numpy.polynomial.legendre as leg
+    >>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0, all real roots
+    array([-0.85099543, -0.11407192,  0.51506735]) # may vary
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) < 2:
+        return np.array([], dtype=c.dtype)
+    if len(c) == 2:
+        return np.array([-c[0] / c[1]])
+
+    # rotated companion matrix reduces error
+    m = legcompanion(c)[::-1, ::-1]
+    r = np.linalg.eigvals(m)
+    r.sort()
+    return r
+
+
+def leggauss(deg):
+    """
+    Gauss-Legendre quadrature.
+
+    Computes the sample points and weights for Gauss-Legendre quadrature.
+    These sample points and weights will correctly integrate polynomials of
+    degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with
+    the weight function :math:`f(x) = 1`.
+
+    Parameters
+    ----------
+    deg : int
+        Number of sample points and weights. It must be >= 1.
+
+    Returns
+    -------
+    x : ndarray
+        1-D ndarray containing the sample points.
+    y : ndarray
+        1-D ndarray containing the weights.
+
+    Notes
+    -----
+    The results have only been tested up to degree 100, higher degrees may
+    be problematic. The weights are determined by using the fact that
+
+    .. math:: w_k = c / (L'_n(x_k) * L_{n-1}(x_k))
+
+    where :math:`c` is a constant independent of :math:`k` and :math:`x_k`
+    is the k'th root of :math:`L_n`, and then scaling the results to get
+    the right value when integrating 1.
+
+    """
+    ideg = pu._as_int(deg, "deg")
+    if ideg <= 0:
+        raise ValueError("deg must be a positive integer")
+
+    # first approximation of roots. We use the fact that the companion
+    # matrix is symmetric in this case in order to obtain better zeros.
+    c = np.array([0] * deg + [1])
+    m = legcompanion(c)
+    x = np.linalg.eigvalsh(m)
+
+    # improve roots by one application of Newton
+    dy = legval(x, c)
+    df = legval(x, legder(c))
+    x -= dy / df
+
+    # compute the weights. We scale the factor to avoid possible numerical
+    # overflow.
+    fm = legval(x, c[1:])
+    fm /= np.abs(fm).max()
+    df /= np.abs(df).max()
+    w = 1 / (fm * df)
+
+    # for Legendre we can also symmetrize
+    w = (w + w[::-1]) / 2
+    x = (x - x[::-1]) / 2
+
+    # scale w to get the right value
+    w *= 2. / w.sum()
+
+    return x, w
+
+
+def legweight(x):
+    """
+    Weight function of the Legendre polynomials.
+
+    The weight function is :math:`1` and the interval of integration is
+    :math:`[-1, 1]`. The Legendre polynomials are orthogonal, but not
+    normalized, with respect to this weight function.
+
+    Parameters
+    ----------
+    x : array_like
+       Values at which the weight function will be computed.
+
+    Returns
+    -------
+    w : ndarray
+       The weight function at `x`.
+    """
+    w = x * 0.0 + 1.0
+    return w
+
+#
+# Legendre series class
+#
+
+class Legendre(ABCPolyBase):
+    """A Legendre series class.
+
+    The Legendre class provides the standard Python numerical methods
+    '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the
+    attributes and methods listed below.
+
+    Parameters
+    ----------
+    coef : array_like
+        Legendre coefficients in order of increasing degree, i.e.,
+        ``(1, 2, 3)`` gives ``1*P_0(x) + 2*P_1(x) + 3*P_2(x)``.
+    domain : (2,) array_like, optional
+        Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
+        to the interval ``[window[0], window[1]]`` by shifting and scaling.
+        The default value is [-1., 1.].
+    window : (2,) array_like, optional
+        Window, see `domain` for its use. The default value is [-1., 1.].
+    symbol : str, optional
+        Symbol used to represent the independent variable in string
+        representations of the polynomial expression, e.g. for printing.
+        The symbol must be a valid Python identifier. Default value is 'x'.
+
+        .. versionadded:: 1.24
+
+    """
+    # Virtual Functions
+    _add = staticmethod(legadd)
+    _sub = staticmethod(legsub)
+    _mul = staticmethod(legmul)
+    _div = staticmethod(legdiv)
+    _pow = staticmethod(legpow)
+    _val = staticmethod(legval)
+    _int = staticmethod(legint)
+    _der = staticmethod(legder)
+    _fit = staticmethod(legfit)
+    _line = staticmethod(legline)
+    _roots = staticmethod(legroots)
+    _fromroots = staticmethod(legfromroots)
+
+    # Virtual properties
+    domain = np.array(legdomain)
+    window = np.array(legdomain)
+    basis_name = 'P'

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy/polynomial/legendre.pyi

@@ -0,0 +1,100 @@
+from typing import Any, ClassVar, Final, Literal as L
+
+import numpy as np
+
+from ._polybase import ABCPolyBase
+from ._polytypes import (
+    _Array1,
+    _Array2,
+    _FuncBinOp,
+    _FuncCompanion,
+    _FuncDer,
+    _FuncFit,
+    _FuncFromRoots,
+    _FuncGauss,
+    _FuncInteg,
+    _FuncLine,
+    _FuncPoly2Ortho,
+    _FuncPow,
+    _FuncRoots,
+    _FuncUnOp,
+    _FuncVal,
+    _FuncVal2D,
+    _FuncVal3D,
+    _FuncVander,
+    _FuncVander2D,
+    _FuncVander3D,
+    _FuncWeight,
+)
+from .polyutils import trimcoef as legtrim
+
+__all__ = [
+    "legzero",
+    "legone",
+    "legx",
+    "legdomain",
+    "legline",
+    "legadd",
+    "legsub",
+    "legmulx",
+    "legmul",
+    "legdiv",
+    "legpow",
+    "legval",
+    "legder",
+    "legint",
+    "leg2poly",
+    "poly2leg",
+    "legfromroots",
+    "legvander",
+    "legfit",
+    "legtrim",
+    "legroots",
+    "Legendre",
+    "legval2d",
+    "legval3d",
+    "leggrid2d",
+    "leggrid3d",
+    "legvander2d",
+    "legvander3d",
+    "legcompanion",
+    "leggauss",
+    "legweight",
+]
+
+poly2leg: Final[_FuncPoly2Ortho] = ...
+leg2poly: Final[_FuncUnOp] = ...
+
+legdomain: Final[_Array2[np.float64]] = ...
+legzero: Final[_Array1[np.int_]] = ...
+legone: Final[_Array1[np.int_]] = ...
+legx: Final[_Array2[np.int_]] = ...
+
+legline: Final[_FuncLine] = ...
+legfromroots: Final[_FuncFromRoots] = ...
+legadd: Final[_FuncBinOp] = ...
+legsub: Final[_FuncBinOp] = ...
+legmulx: Final[_FuncUnOp] = ...
+legmul: Final[_FuncBinOp] = ...
+legdiv: Final[_FuncBinOp] = ...
+legpow: Final[_FuncPow] = ...
+legder: Final[_FuncDer] = ...
+legint: Final[_FuncInteg] = ...
+legval: Final[_FuncVal] = ...
+legval2d: Final[_FuncVal2D] = ...
+legval3d: Final[_FuncVal3D] = ...
+leggrid2d: Final[_FuncVal2D] = ...
+leggrid3d: Final[_FuncVal3D] = ...
+legvander: Final[_FuncVander] = ...
+legvander2d: Final[_FuncVander2D] = ...
+legvander3d: Final[_FuncVander3D] = ...
+legfit: Final[_FuncFit] = ...
+legcompanion: Final[_FuncCompanion] = ...
+legroots: Final[_FuncRoots] = ...
+leggauss: Final[_FuncGauss] = ...
+legweight: Final[_FuncWeight] = ...
+
+class Legendre(ABCPolyBase[L["P"]]):
+    basis_name: ClassVar[L["P"]] = "P"  # pyright: ignore[reportIncompatibleMethodOverride]
+    domain: _Array2[np.float64 | Any] = ...  # pyright: ignore[reportIncompatibleMethodOverride]
+    window: _Array2[np.float64 | Any] = ...  # pyright: ignore[reportIncompatibleMethodOverride]

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy/polynomial/polynomial.py

@@ -0,0 +1,1625 @@
+"""
+=================================================
+Power Series (:mod:`numpy.polynomial.polynomial`)
+=================================================
+
+This module provides a number of objects (mostly functions) useful for
+dealing with polynomials, including a `Polynomial` class that
+encapsulates the usual arithmetic operations.  (General information
+on how this module represents and works with polynomial objects is in
+the docstring for its "parent" sub-package, `numpy.polynomial`).
+
+Classes
+-------
+.. autosummary::
+   :toctree: generated/
+
+   Polynomial
+
+Constants
+---------
+.. autosummary::
+   :toctree: generated/
+
+   polydomain
+   polyzero
+   polyone
+   polyx
+
+Arithmetic
+----------
+.. autosummary::
+   :toctree: generated/
+
+   polyadd
+   polysub
+   polymulx
+   polymul
+   polydiv
+   polypow
+   polyval
+   polyval2d
+   polyval3d
+   polygrid2d
+   polygrid3d
+
+Calculus
+--------
+.. autosummary::
+   :toctree: generated/
+
+   polyder
+   polyint
+
+Misc Functions
+--------------
+.. autosummary::
+   :toctree: generated/
+
+   polyfromroots
+   polyroots
+   polyvalfromroots
+   polyvander
+   polyvander2d
+   polyvander3d
+   polycompanion
+   polyfit
+   polytrim
+   polyline
+
+See Also
+--------
+`numpy.polynomial`
+
+"""
+__all__ = [
+    'polyzero', 'polyone', 'polyx', 'polydomain', 'polyline', 'polyadd',
+    'polysub', 'polymulx', 'polymul', 'polydiv', 'polypow', 'polyval',
+    'polyvalfromroots', 'polyder', 'polyint', 'polyfromroots', 'polyvander',
+    'polyfit', 'polytrim', 'polyroots', 'Polynomial', 'polyval2d', 'polyval3d',
+    'polygrid2d', 'polygrid3d', 'polyvander2d', 'polyvander3d',
+    'polycompanion']
+
+import numpy as np
+from numpy._core.overrides import array_function_dispatch as _array_function_dispatch
+
+from . import polyutils as pu
+from ._polybase import ABCPolyBase
+
+polytrim = pu.trimcoef
+
+#
+# These are constant arrays are of integer type so as to be compatible
+# with the widest range of other types, such as Decimal.
+#
+
+# Polynomial default domain.
+polydomain = np.array([-1., 1.])
+
+# Polynomial coefficients representing zero.
+polyzero = np.array([0])
+
+# Polynomial coefficients representing one.
+polyone = np.array([1])
+
+# Polynomial coefficients representing the identity x.
+polyx = np.array([0, 1])
+
+#
+# Polynomial series functions
+#
+
+
+def polyline(off, scl):
+    """
+    Returns an array representing a linear polynomial.
+
+    Parameters
+    ----------
+    off, scl : scalars
+        The "y-intercept" and "slope" of the line, respectively.
+
+    Returns
+    -------
+    y : ndarray
+        This module's representation of the linear polynomial ``off +
+        scl*x``.
+
+    See Also
+    --------
+    numpy.polynomial.chebyshev.chebline
+    numpy.polynomial.legendre.legline
+    numpy.polynomial.laguerre.lagline
+    numpy.polynomial.hermite.hermline
+    numpy.polynomial.hermite_e.hermeline
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> P.polyline(1, -1)
+    array([ 1, -1])
+    >>> P.polyval(1, P.polyline(1, -1))  # should be 0
+    0.0
+
+    """
+    if scl != 0:
+        return np.array([off, scl])
+    else:
+        return np.array([off])
+
+
+def polyfromroots(roots):
+    """
+    Generate a monic polynomial with given roots.
+
+    Return the coefficients of the polynomial
+
+    .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
+
+    where the :math:`r_n` are the roots specified in `roots`.  If a zero has
+    multiplicity n, then it must appear in `roots` n times. For instance,
+    if 2 is a root of multiplicity three and 3 is a root of multiplicity 2,
+    then `roots` looks something like [2, 2, 2, 3, 3]. The roots can appear
+    in any order.
+
+    If the returned coefficients are `c`, then
+
+    .. math:: p(x) = c_0 + c_1 * x + ... +  x^n
+
+    The coefficient of the last term is 1 for monic polynomials in this
+    form.
+
+    Parameters
+    ----------
+    roots : array_like
+        Sequence containing the roots.
+
+    Returns
+    -------
+    out : ndarray
+        1-D array of the polynomial's coefficients If all the roots are
+        real, then `out` is also real, otherwise it is complex.  (see
+        Examples below).
+
+    See Also
+    --------
+    numpy.polynomial.chebyshev.chebfromroots
+    numpy.polynomial.legendre.legfromroots
+    numpy.polynomial.laguerre.lagfromroots
+    numpy.polynomial.hermite.hermfromroots
+    numpy.polynomial.hermite_e.hermefromroots
+
+    Notes
+    -----
+    The coefficients are determined by multiplying together linear factors
+    of the form ``(x - r_i)``, i.e.
+
+    .. math:: p(x) = (x - r_0) (x - r_1) ... (x - r_n)
+
+    where ``n == len(roots) - 1``; note that this implies that ``1`` is always
+    returned for :math:`a_n`.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> P.polyfromroots((-1,0,1))  # x(x - 1)(x + 1) = x^3 - x
+    array([ 0., -1.,  0.,  1.])
+    >>> j = complex(0,1)
+    >>> P.polyfromroots((-j,j))  # complex returned, though values are real
+    array([1.+0.j,  0.+0.j,  1.+0.j])
+
+    """
+    return pu._fromroots(polyline, polymul, roots)
+
+
+def polyadd(c1, c2):
+    """
+    Add one polynomial to another.
+
+    Returns the sum of two polynomials `c1` + `c2`.  The arguments are
+    sequences of coefficients from lowest order term to highest, i.e.,
+    [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of polynomial coefficients ordered from low to high.
+
+    Returns
+    -------
+    out : ndarray
+        The coefficient array representing their sum.
+
+    See Also
+    --------
+    polysub, polymulx, polymul, polydiv, polypow
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> c1 = (1, 2, 3)
+    >>> c2 = (3, 2, 1)
+    >>> sum = P.polyadd(c1,c2); sum
+    array([4.,  4.,  4.])
+    >>> P.polyval(2, sum)  # 4 + 4(2) + 4(2**2)
+    28.0
+
+    """
+    return pu._add(c1, c2)
+
+
+def polysub(c1, c2):
+    """
+    Subtract one polynomial from another.
+
+    Returns the difference of two polynomials `c1` - `c2`.  The arguments
+    are sequences of coefficients from lowest order term to highest, i.e.,
+    [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of polynomial coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of coefficients representing their difference.
+
+    See Also
+    --------
+    polyadd, polymulx, polymul, polydiv, polypow
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> c1 = (1, 2, 3)
+    >>> c2 = (3, 2, 1)
+    >>> P.polysub(c1,c2)
+    array([-2.,  0.,  2.])
+    >>> P.polysub(c2, c1)  # -P.polysub(c1,c2)
+    array([ 2.,  0., -2.])
+
+    """
+    return pu._sub(c1, c2)
+
+
+def polymulx(c):
+    """Multiply a polynomial by x.
+
+    Multiply the polynomial `c` by x, where x is the independent
+    variable.
+
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of polynomial coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the result of the multiplication.
+
+    See Also
+    --------
+    polyadd, polysub, polymul, polydiv, polypow
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> c = (1, 2, 3)
+    >>> P.polymulx(c)
+    array([0., 1., 2., 3.])
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    # The zero series needs special treatment
+    if len(c) == 1 and c[0] == 0:
+        return c
+
+    prd = np.empty(len(c) + 1, dtype=c.dtype)
+    prd[0] = c[0] * 0
+    prd[1:] = c
+    return prd
+
+
+def polymul(c1, c2):
+    """
+    Multiply one polynomial by another.
+
+    Returns the product of two polynomials `c1` * `c2`.  The arguments are
+    sequences of coefficients, from lowest order term to highest, e.g.,
+    [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2.``
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of coefficients representing a polynomial, relative to the
+        "standard" basis, and ordered from lowest order term to highest.
+
+    Returns
+    -------
+    out : ndarray
+        Of the coefficients of their product.
+
+    See Also
+    --------
+    polyadd, polysub, polymulx, polydiv, polypow
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> c1 = (1, 2, 3)
+    >>> c2 = (3, 2, 1)
+    >>> P.polymul(c1, c2)
+    array([  3.,   8.,  14.,   8.,   3.])
+
+    """
+    # c1, c2 are trimmed copies
+    [c1, c2] = pu.as_series([c1, c2])
+    ret = np.convolve(c1, c2)
+    return pu.trimseq(ret)
+
+
+def polydiv(c1, c2):
+    """
+    Divide one polynomial by another.
+
+    Returns the quotient-with-remainder of two polynomials `c1` / `c2`.
+    The arguments are sequences of coefficients, from lowest order term
+    to highest, e.g., [1,2,3] represents ``1 + 2*x + 3*x**2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of polynomial coefficients ordered from low to high.
+
+    Returns
+    -------
+    [quo, rem] : ndarrays
+        Of coefficient series representing the quotient and remainder.
+
+    See Also
+    --------
+    polyadd, polysub, polymulx, polymul, polypow
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> c1 = (1, 2, 3)
+    >>> c2 = (3, 2, 1)
+    >>> P.polydiv(c1, c2)
+    (array([3.]), array([-8., -4.]))
+    >>> P.polydiv(c2, c1)
+    (array([ 0.33333333]), array([ 2.66666667,  1.33333333]))  # may vary
+
+    """
+    # c1, c2 are trimmed copies
+    [c1, c2] = pu.as_series([c1, c2])
+    if c2[-1] == 0:
+        raise ZeroDivisionError  # FIXME: add message with details to exception
+
+    # note: this is more efficient than `pu._div(polymul, c1, c2)`
+    lc1 = len(c1)
+    lc2 = len(c2)
+    if lc1 < lc2:
+        return c1[:1] * 0, c1
+    elif lc2 == 1:
+        return c1 / c2[-1], c1[:1] * 0
+    else:
+        dlen = lc1 - lc2
+        scl = c2[-1]
+        c2 = c2[:-1] / scl
+        i = dlen
+        j = lc1 - 1
+        while i >= 0:
+            c1[i:j] -= c2 * c1[j]
+            i -= 1
+            j -= 1
+        return c1[j + 1:] / scl, pu.trimseq(c1[:j + 1])
+
+
+def polypow(c, pow, maxpower=None):
+    """Raise a polynomial to a power.
+
+    Returns the polynomial `c` raised to the power `pow`. The argument
+    `c` is a sequence of coefficients ordered from low to high. i.e.,
+    [1,2,3] is the series  ``1 + 2*x + 3*x**2.``
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of array of series coefficients ordered from low to
+        high degree.
+    pow : integer
+        Power to which the series will be raised
+    maxpower : integer, optional
+        Maximum power allowed. This is mainly to limit growth of the series
+        to unmanageable size. Default is 16
+
+    Returns
+    -------
+    coef : ndarray
+        Power series of power.
+
+    See Also
+    --------
+    polyadd, polysub, polymulx, polymul, polydiv
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> P.polypow([1, 2, 3], 2)
+    array([ 1., 4., 10., 12., 9.])
+
+    """
+    # note: this is more efficient than `pu._pow(polymul, c1, c2)`, as it
+    # avoids calling `as_series` repeatedly
+    return pu._pow(np.convolve, c, pow, maxpower)
+
+
+def polyder(c, m=1, scl=1, axis=0):
+    """
+    Differentiate a polynomial.
+
+    Returns the polynomial coefficients `c` differentiated `m` times along
+    `axis`.  At each iteration the result is multiplied by `scl` (the
+    scaling factor is for use in a linear change of variable).  The
+    argument `c` is an array of coefficients from low to high degree along
+    each axis, e.g., [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``
+    while [[1,2],[1,2]] represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is
+    ``x`` and axis=1 is ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of polynomial coefficients. If c is multidimensional the
+        different axis correspond to different variables with the degree
+        in each axis given by the corresponding index.
+    m : int, optional
+        Number of derivatives taken, must be non-negative. (Default: 1)
+    scl : scalar, optional
+        Each differentiation is multiplied by `scl`.  The end result is
+        multiplication by ``scl**m``.  This is for use in a linear change
+        of variable. (Default: 1)
+    axis : int, optional
+        Axis over which the derivative is taken. (Default: 0).
+
+    Returns
+    -------
+    der : ndarray
+        Polynomial coefficients of the derivative.
+
+    See Also
+    --------
+    polyint
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> c = (1, 2, 3, 4)
+    >>> P.polyder(c)  # (d/dx)(c)
+    array([  2.,   6.,  12.])
+    >>> P.polyder(c, 3)  # (d**3/dx**3)(c)
+    array([24.])
+    >>> P.polyder(c, scl=-1)  # (d/d(-x))(c)
+    array([ -2.,  -6., -12.])
+    >>> P.polyder(c, 2, -1)  # (d**2/d(-x)**2)(c)
+    array([  6.,  24.])
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        # astype fails with NA
+        c = c + 0.0
+    cdt = c.dtype
+    cnt = pu._as_int(m, "the order of derivation")
+    iaxis = pu._as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of derivation must be non-negative")
+    iaxis = np.lib.array_utils.normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    n = len(c)
+    if cnt >= n:
+        c = c[:1] * 0
+    else:
+        for i in range(cnt):
+            n = n - 1
+            c *= scl
+            der = np.empty((n,) + c.shape[1:], dtype=cdt)
+            for j in range(n, 0, -1):
+                der[j - 1] = j * c[j]
+            c = der
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def polyint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
+    """
+    Integrate a polynomial.
+
+    Returns the polynomial coefficients `c` integrated `m` times from
+    `lbnd` along `axis`.  At each iteration the resulting series is
+    **multiplied** by `scl` and an integration constant, `k`, is added.
+    The scaling factor is for use in a linear change of variable.  ("Buyer
+    beware": note that, depending on what one is doing, one may want `scl`
+    to be the reciprocal of what one might expect; for more information,
+    see the Notes section below.) The argument `c` is an array of
+    coefficients, from low to high degree along each axis, e.g., [1,2,3]
+    represents the polynomial ``1 + 2*x + 3*x**2`` while [[1,2],[1,2]]
+    represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is ``x`` and axis=1 is
+    ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of polynomial coefficients, ordered from low to high.
+    m : int, optional
+        Order of integration, must be positive. (Default: 1)
+    k : {[], list, scalar}, optional
+        Integration constant(s).  The value of the first integral at zero
+        is the first value in the list, the value of the second integral
+        at zero is the second value, etc.  If ``k == []`` (the default),
+        all constants are set to zero.  If ``m == 1``, a single scalar can
+        be given instead of a list.
+    lbnd : scalar, optional
+        The lower bound of the integral. (Default: 0)
+    scl : scalar, optional
+        Following each integration the result is *multiplied* by `scl`
+        before the integration constant is added. (Default: 1)
+    axis : int, optional
+        Axis over which the integral is taken. (Default: 0).
+
+    Returns
+    -------
+    S : ndarray
+        Coefficient array of the integral.
+
+    Raises
+    ------
+    ValueError
+        If ``m < 1``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
+        ``np.ndim(scl) != 0``.
+
+    See Also
+    --------
+    polyder
+
+    Notes
+    -----
+    Note that the result of each integration is *multiplied* by `scl`.  Why
+    is this important to note?  Say one is making a linear change of
+    variable :math:`u = ax + b` in an integral relative to `x`. Then
+    :math:`dx = du/a`, so one will need to set `scl` equal to
+    :math:`1/a` - perhaps not what one would have first thought.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> c = (1, 2, 3)
+    >>> P.polyint(c)  # should return array([0, 1, 1, 1])
+    array([0.,  1.,  1.,  1.])
+    >>> P.polyint(c, 3)  # should return array([0, 0, 0, 1/6, 1/12, 1/20])
+     array([ 0.        ,  0.        ,  0.        ,  0.16666667,  0.08333333, # may vary
+             0.05      ])
+    >>> P.polyint(c, k=3)  # should return array([3, 1, 1, 1])
+    array([3.,  1.,  1.,  1.])
+    >>> P.polyint(c,lbnd=-2)  # should return array([6, 1, 1, 1])
+    array([6.,  1.,  1.,  1.])
+    >>> P.polyint(c,scl=-2)  # should return array([0, -2, -2, -2])
+    array([ 0., -2., -2., -2.])
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        # astype doesn't preserve mask attribute.
+        c = c + 0.0
+    cdt = c.dtype
+    if not np.iterable(k):
+        k = [k]
+    cnt = pu._as_int(m, "the order of integration")
+    iaxis = pu._as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of integration must be non-negative")
+    if len(k) > cnt:
+        raise ValueError("Too many integration constants")
+    if np.ndim(lbnd) != 0:
+        raise ValueError("lbnd must be a scalar.")
+    if np.ndim(scl) != 0:
+        raise ValueError("scl must be a scalar.")
+    iaxis = np.lib.array_utils.normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    k = list(k) + [0] * (cnt - len(k))
+    c = np.moveaxis(c, iaxis, 0)
+    for i in range(cnt):
+        n = len(c)
+        c *= scl
+        if n == 1 and np.all(c[0] == 0):
+            c[0] += k[i]
+        else:
+            tmp = np.empty((n + 1,) + c.shape[1:], dtype=cdt)
+            tmp[0] = c[0] * 0
+            tmp[1] = c[0]
+            for j in range(1, n):
+                tmp[j + 1] = c[j] / (j + 1)
+            tmp[0] += k[i] - polyval(lbnd, tmp)
+            c = tmp
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def polyval(x, c, tensor=True):
+    """
+    Evaluate a polynomial at points x.
+
+    If `c` is of length ``n + 1``, this function returns the value
+
+    .. math:: p(x) = c_0 + c_1 * x + ... + c_n * x^n
+
+    The parameter `x` is converted to an array only if it is a tuple or a
+    list, otherwise it is treated as a scalar. In either case, either `x`
+    or its elements must support multiplication and addition both with
+    themselves and with the elements of `c`.
+
+    If `c` is a 1-D array, then ``p(x)`` will have the same shape as `x`.  If
+    `c` is multidimensional, then the shape of the result depends on the
+    value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
+    x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
+    scalars have shape (,).
+
+    Trailing zeros in the coefficients will be used in the evaluation, so
+    they should be avoided if efficiency is a concern.
+
+    Parameters
+    ----------
+    x : array_like, compatible object
+        If `x` is a list or tuple, it is converted to an ndarray, otherwise
+        it is left unchanged and treated as a scalar. In either case, `x`
+        or its elements must support addition and multiplication with
+        with themselves and with the elements of `c`.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree n are contained in c[n]. If `c` is multidimensional the
+        remaining indices enumerate multiple polynomials. In the two
+        dimensional case the coefficients may be thought of as stored in
+        the columns of `c`.
+    tensor : boolean, optional
+        If True, the shape of the coefficient array is extended with ones
+        on the right, one for each dimension of `x`. Scalars have dimension 0
+        for this action. The result is that every column of coefficients in
+        `c` is evaluated for every element of `x`. If False, `x` is broadcast
+        over the columns of `c` for the evaluation.  This keyword is useful
+        when `c` is multidimensional. The default value is True.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The shape of the returned array is described above.
+
+    See Also
+    --------
+    polyval2d, polygrid2d, polyval3d, polygrid3d
+
+    Notes
+    -----
+    The evaluation uses Horner's method.
+
+    When using coefficients from polynomials created with ``Polynomial.fit()``,
+    use ``p(x)`` or ``polyval(x, p.convert().coef)`` to handle domain/window
+    scaling correctly, not ``polyval(x, p.coef)``.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from numpy.polynomial.polynomial import polyval
+    >>> polyval(1, [1,2,3])
+    6.0
+    >>> a = np.arange(4).reshape(2,2)
+    >>> a
+    array([[0, 1],
+           [2, 3]])
+    >>> polyval(a, [1, 2, 3])
+    array([[ 1.,   6.],
+           [17.,  34.]])
+    >>> coef = np.arange(4).reshape(2, 2)  # multidimensional coefficients
+    >>> coef
+    array([[0, 1],
+           [2, 3]])
+    >>> polyval([1, 2], coef, tensor=True)
+    array([[2.,  4.],
+           [4.,  7.]])
+    >>> polyval([1, 2], coef, tensor=False)
+    array([2.,  7.])
+
+    """
+    c = np.array(c, ndmin=1, copy=None)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        # astype fails with NA
+        c = c + 0.0
+    if isinstance(x, (tuple, list)):
+        x = np.asarray(x)
+    if isinstance(x, np.ndarray) and tensor:
+        c = c.reshape(c.shape + (1,) * x.ndim)
+
+    c0 = c[-1] + x * 0
+    for i in range(2, len(c) + 1):
+        c0 = c[-i] + c0 * x
+    return c0
+
+
+def polyvalfromroots(x, r, tensor=True):
+    """
+    Evaluate a polynomial specified by its roots at points x.
+
+    If `r` is of length ``N``, this function returns the value
+
+    .. math:: p(x) = \\prod_{n=1}^{N} (x - r_n)
+
+    The parameter `x` is converted to an array only if it is a tuple or a
+    list, otherwise it is treated as a scalar. In either case, either `x`
+    or its elements must support multiplication and addition both with
+    themselves and with the elements of `r`.
+
+    If `r` is a 1-D array, then ``p(x)`` will have the same shape as `x`.  If `r`
+    is multidimensional, then the shape of the result depends on the value of
+    `tensor`. If `tensor` is ``True`` the shape will be r.shape[1:] + x.shape;
+    that is, each polynomial is evaluated at every value of `x`. If `tensor` is
+    ``False``, the shape will be r.shape[1:]; that is, each polynomial is
+    evaluated only for the corresponding broadcast value of `x`. Note that
+    scalars have shape (,).
+
+    Parameters
+    ----------
+    x : array_like, compatible object
+        If `x` is a list or tuple, it is converted to an ndarray, otherwise
+        it is left unchanged and treated as a scalar. In either case, `x`
+        or its elements must support addition and multiplication with
+        with themselves and with the elements of `r`.
+    r : array_like
+        Array of roots. If `r` is multidimensional the first index is the
+        root index, while the remaining indices enumerate multiple
+        polynomials. For instance, in the two dimensional case the roots
+        of each polynomial may be thought of as stored in the columns of `r`.
+    tensor : boolean, optional
+        If True, the shape of the roots array is extended with ones on the
+        right, one for each dimension of `x`. Scalars have dimension 0 for this
+        action. The result is that every column of coefficients in `r` is
+        evaluated for every element of `x`. If False, `x` is broadcast over the
+        columns of `r` for the evaluation.  This keyword is useful when `r` is
+        multidimensional. The default value is True.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The shape of the returned array is described above.
+
+    See Also
+    --------
+    polyroots, polyfromroots, polyval
+
+    Examples
+    --------
+    >>> from numpy.polynomial.polynomial import polyvalfromroots
+    >>> polyvalfromroots(1, [1, 2, 3])
+    0.0
+    >>> a = np.arange(4).reshape(2, 2)
+    >>> a
+    array([[0, 1],
+           [2, 3]])
+    >>> polyvalfromroots(a, [-1, 0, 1])
+    array([[-0.,   0.],
+           [ 6.,  24.]])
+    >>> r = np.arange(-2, 2).reshape(2,2)  # multidimensional coefficients
+    >>> r # each column of r defines one polynomial
+    array([[-2, -1],
+           [ 0,  1]])
+    >>> b = [-2, 1]
+    >>> polyvalfromroots(b, r, tensor=True)
+    array([[-0.,  3.],
+           [ 3., 0.]])
+    >>> polyvalfromroots(b, r, tensor=False)
+    array([-0.,  0.])
+
+    """
+    r = np.array(r, ndmin=1, copy=None)
+    if r.dtype.char in '?bBhHiIlLqQpP':
+        r = r.astype(np.double)
+    if isinstance(x, (tuple, list)):
+        x = np.asarray(x)
+    if isinstance(x, np.ndarray):
+        if tensor:
+            r = r.reshape(r.shape + (1,) * x.ndim)
+        elif x.ndim >= r.ndim:
+            raise ValueError("x.ndim must be < r.ndim when tensor == False")
+    return np.prod(x - r, axis=0)
+
+def _polyval2d_dispatcher(x, y, c):
+    return (x, y, c)
+
+def _polygrid2d_dispatcher(x, y, c):
+    return (x, y, c)
+
+@_array_function_dispatch(_polyval2d_dispatcher)
+def polyval2d(x, y, c):
+    """
+    Evaluate a 2-D polynomial at points (x, y).
+
+    This function returns the value
+
+    .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * x^i * y^j
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars and they
+    must have the same shape after conversion. In either case, either `x`
+    and `y` or their elements must support multiplication and addition both
+    with themselves and with the elements of `c`.
+
+    If `c` has fewer than two dimensions, ones are implicitly appended to
+    its shape to make it 2-D. The shape of the result will be c.shape[2:] +
+    x.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points ``(x, y)``,
+        where `x` and `y` must have the same shape. If `x` or `y` is a list
+        or tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and, if it isn't an ndarray, it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term
+        of multi-degree i,j is contained in ``c[i,j]``. If `c` has
+        dimension greater than two the remaining indices enumerate multiple
+        sets of coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points formed with
+        pairs of corresponding values from `x` and `y`.
+
+    See Also
+    --------
+    polyval, polygrid2d, polyval3d, polygrid3d
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> c = ((1, 2, 3), (4, 5, 6))
+    >>> P.polyval2d(1, 1, c)
+    21.0
+
+    """
+    return pu._valnd(polyval, c, x, y)
+
+@_array_function_dispatch(_polygrid2d_dispatcher)
+def polygrid2d(x, y, c):
+    """
+    Evaluate a 2-D polynomial on the Cartesian product of x and y.
+
+    This function returns the values:
+
+    .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * a^i * b^j
+
+    where the points ``(a, b)`` consist of all pairs formed by taking
+    `a` from `x` and `b` from `y`. The resulting points form a grid with
+    `x` in the first dimension and `y` in the second.
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars. In either
+    case, either `x` and `y` or their elements must support multiplication
+    and addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than two dimensions, ones are implicitly appended to
+    its shape to make it 2-D. The shape of the result will be c.shape[2:] +
+    x.shape + y.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points in the
+        Cartesian product of `x` and `y`.  If `x` or `y` is a list or
+        tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and, if it isn't an ndarray, it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree i,j are contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points in the Cartesian
+        product of `x` and `y`.
+
+    See Also
+    --------
+    polyval, polyval2d, polyval3d, polygrid3d
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> c = ((1, 2, 3), (4, 5, 6))
+    >>> P.polygrid2d([0, 1], [0, 1], c)
+    array([[ 1.,  6.],
+           [ 5., 21.]])
+
+    """
+    return pu._gridnd(polyval, c, x, y)
+
+
+def polyval3d(x, y, z, c):
+    """
+    Evaluate a 3-D polynomial at points (x, y, z).
+
+    This function returns the values:
+
+    .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * x^i * y^j * z^k
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if
+    they are tuples or a lists, otherwise they are treated as a scalars and
+    they must have the same shape after conversion. In either case, either
+    `x`, `y`, and `z` or their elements must support multiplication and
+    addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than 3 dimensions, ones are implicitly appended to its
+    shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible object
+        The three dimensional series is evaluated at the points
+        ``(x, y, z)``, where `x`, `y`, and `z` must have the same shape.  If
+        any of `x`, `y`, or `z` is a list or tuple, it is first converted
+        to an ndarray, otherwise it is left unchanged and if it isn't an
+        ndarray it is  treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term of
+        multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
+        greater than 3 the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the multidimensional polynomial on points formed with
+        triples of corresponding values from `x`, `y`, and `z`.
+
+    See Also
+    --------
+    polyval, polyval2d, polygrid2d, polygrid3d
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> c = ((1, 2, 3), (4, 5, 6), (7, 8, 9))
+    >>> P.polyval3d(1, 1, 1, c)
+    45.0
+
+    """
+    return pu._valnd(polyval, c, x, y, z)
+
+
+def polygrid3d(x, y, z, c):
+    """
+    Evaluate a 3-D polynomial on the Cartesian product of x, y and z.
+
+    This function returns the values:
+
+    .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * a^i * b^j * c^k
+
+    where the points ``(a, b, c)`` consist of all triples formed by taking
+    `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
+    a grid with `x` in the first dimension, `y` in the second, and `z` in
+    the third.
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if they
+    are tuples or a lists, otherwise they are treated as a scalars. In
+    either case, either `x`, `y`, and `z` or their elements must support
+    multiplication and addition both with themselves and with the elements
+    of `c`.
+
+    If `c` has fewer than three dimensions, ones are implicitly appended to
+    its shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape + y.shape + z.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible objects
+        The three dimensional series is evaluated at the points in the
+        Cartesian product of `x`, `y`, and `z`.  If `x`, `y`, or `z` is a
+        list or tuple, it is first converted to an ndarray, otherwise it is
+        left unchanged and, if it isn't an ndarray, it is treated as a
+        scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree i,j are contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points in the Cartesian
+        product of `x` and `y`.
+
+    See Also
+    --------
+    polyval, polyval2d, polygrid2d, polyval3d
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> c = ((1, 2, 3), (4, 5, 6), (7, 8, 9))
+    >>> P.polygrid3d([0, 1], [0, 1], [0, 1], c)
+    array([[ 1., 13.],
+           [ 6., 51.]])
+
+    """
+    return pu._gridnd(polyval, c, x, y, z)
+
+
+def polyvander(x, deg):
+    """Vandermonde matrix of given degree.
+
+    Returns the Vandermonde matrix of degree `deg` and sample points
+    `x`. The Vandermonde matrix is defined by
+
+    .. math:: V[..., i] = x^i,
+
+    where ``0 <= i <= deg``. The leading indices of `V` index the elements of
+    `x` and the last index is the power of `x`.
+
+    If `c` is a 1-D array of coefficients of length ``n + 1`` and `V` is the
+    matrix ``V = polyvander(x, n)``, then ``np.dot(V, c)`` and
+    ``polyval(x, c)`` are the same up to roundoff. This equivalence is
+    useful both for least squares fitting and for the evaluation of a large
+    number of polynomials of the same degree and sample points.
+
+    Parameters
+    ----------
+    x : array_like
+        Array of points. The dtype is converted to float64 or complex128
+        depending on whether any of the elements are complex. If `x` is
+        scalar it is converted to a 1-D array.
+    deg : int
+        Degree of the resulting matrix.
+
+    Returns
+    -------
+    vander : ndarray.
+        The Vandermonde matrix. The shape of the returned matrix is
+        ``x.shape + (deg + 1,)``, where the last index is the power of `x`.
+        The dtype will be the same as the converted `x`.
+
+    See Also
+    --------
+    polyvander2d, polyvander3d
+
+    Examples
+    --------
+    The Vandermonde matrix of degree ``deg = 5`` and sample points
+    ``x = [-1, 2, 3]`` contains the element-wise powers of `x`
+    from 0 to 5 as its columns.
+
+    >>> from numpy.polynomial import polynomial as P
+    >>> x, deg = [-1, 2, 3], 5
+    >>> P.polyvander(x=x, deg=deg)
+    array([[  1.,  -1.,   1.,  -1.,   1.,  -1.],
+           [  1.,   2.,   4.,   8.,  16.,  32.],
+           [  1.,   3.,   9.,  27.,  81., 243.]])
+
+    """
+    ideg = pu._as_int(deg, "deg")
+    if ideg < 0:
+        raise ValueError("deg must be non-negative")
+
+    x = np.array(x, copy=None, ndmin=1) + 0.0
+    dims = (ideg + 1,) + x.shape
+    dtyp = x.dtype
+    v = np.empty(dims, dtype=dtyp)
+    v[0] = x * 0 + 1
+    if ideg > 0:
+        v[1] = x
+        for i in range(2, ideg + 1):
+            v[i] = v[i - 1] * x
+    return np.moveaxis(v, 0, -1)
+
+
+def polyvander2d(x, y, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points ``(x, y)``. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (deg[1] + 1)*i + j] = x^i * y^j,
+
+    where ``0 <= i <= deg[0]`` and ``0 <= j <= deg[1]``. The leading indices of
+    `V` index the points ``(x, y)`` and the last index encodes the powers of
+    `x` and `y`.
+
+    If ``V = polyvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
+    correspond to the elements of a 2-D coefficient array `c` of shape
+    (xdeg + 1, ydeg + 1) in the order
+
+    .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
+
+    and ``np.dot(V, c.flat)`` and ``polyval2d(x, y, c)`` will be the same
+    up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 2-D polynomials
+    of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes
+        will be converted to either float64 or complex128 depending on
+        whether any of the elements are complex. Scalars are converted to
+        1-D arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg].
+
+    Returns
+    -------
+    vander2d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg([1]+1)`.  The dtype will be the same
+        as the converted `x` and `y`.
+
+    See Also
+    --------
+    polyvander, polyvander3d, polyval2d, polyval3d
+
+    Examples
+    --------
+    >>> import numpy as np
+
+    The 2-D pseudo-Vandermonde matrix of degree ``[1, 2]`` and sample
+    points ``x = [-1, 2]`` and ``y = [1, 3]`` is as follows:
+
+    >>> from numpy.polynomial import polynomial as P
+    >>> x = np.array([-1, 2])
+    >>> y = np.array([1, 3])
+    >>> m, n = 1, 2
+    >>> deg = np.array([m, n])
+    >>> V = P.polyvander2d(x=x, y=y, deg=deg)
+    >>> V
+    array([[ 1.,  1.,  1., -1., -1., -1.],
+           [ 1.,  3.,  9.,  2.,  6., 18.]])
+
+    We can verify the columns for any ``0 <= i <= m`` and ``0 <= j <= n``:
+
+    >>> i, j = 0, 1
+    >>> V[:, (deg[1]+1)*i + j] == x**i * y**j
+    array([ True,  True])
+
+    The (1D) Vandermonde matrix of sample points ``x`` and degree ``m`` is a
+    special case of the (2D) pseudo-Vandermonde matrix with ``y`` points all
+    zero and degree ``[m, 0]``.
+
+    >>> P.polyvander2d(x=x, y=0*x, deg=(m, 0)) == P.polyvander(x=x, deg=m)
+    array([[ True,  True],
+           [ True,  True]])
+
+    """
+    return pu._vander_nd_flat((polyvander, polyvander), (x, y), deg)
+
+
+def polyvander3d(x, y, z, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points ``(x, y, z)``. If `l`, `m`, `n` are the given degrees in `x`, `y`, `z`,
+    then The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = x^i * y^j * z^k,
+
+    where ``0 <= i <= l``, ``0 <= j <= m``, and ``0 <= j <= n``.  The leading
+    indices of `V` index the points ``(x, y, z)`` and the last index encodes
+    the powers of `x`, `y`, and `z`.
+
+    If ``V = polyvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
+    of `V` correspond to the elements of a 3-D coefficient array `c` of
+    shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
+
+    .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
+
+    and  ``np.dot(V, c.flat)`` and ``polyval3d(x, y, z, c)`` will be the
+    same up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 3-D polynomials
+    of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y, z : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes will
+        be converted to either float64 or complex128 depending on whether
+        any of the elements are complex. Scalars are converted to 1-D
+        arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg, z_deg].
+
+    Returns
+    -------
+    vander3d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg([1]+1)*(deg[2]+1)`.  The dtype will
+        be the same as the converted `x`, `y`, and `z`.
+
+    See Also
+    --------
+    polyvander, polyvander3d, polyval2d, polyval3d
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from numpy.polynomial import polynomial as P
+    >>> x = np.asarray([-1, 2, 1])
+    >>> y = np.asarray([1, -2, -3])
+    >>> z = np.asarray([2, 2, 5])
+    >>> l, m, n = [2, 2, 1]
+    >>> deg = [l, m, n]
+    >>> V = P.polyvander3d(x=x, y=y, z=z, deg=deg)
+    >>> V
+    array([[  1.,   2.,   1.,   2.,   1.,   2.,  -1.,  -2.,  -1.,
+             -2.,  -1.,  -2.,   1.,   2.,   1.,   2.,   1.,   2.],
+           [  1.,   2.,  -2.,  -4.,   4.,   8.,   2.,   4.,  -4.,
+             -8.,   8.,  16.,   4.,   8.,  -8., -16.,  16.,  32.],
+           [  1.,   5.,  -3., -15.,   9.,  45.,   1.,   5.,  -3.,
+            -15.,   9.,  45.,   1.,   5.,  -3., -15.,   9.,  45.]])
+
+    We can verify the columns for any ``0 <= i <= l``, ``0 <= j <= m``,
+    and ``0 <= k <= n``
+
+    >>> i, j, k = 2, 1, 0
+    >>> V[:, (m+1)*(n+1)*i + (n+1)*j + k] == x**i * y**j * z**k
+    array([ True,  True,  True])
+
+    """
+    return pu._vander_nd_flat((polyvander, polyvander, polyvander), (x, y, z), deg)
+
+
+def polyfit(x, y, deg, rcond=None, full=False, w=None):
+    """
+    Least-squares fit of a polynomial to data.
+
+    Return the coefficients of a polynomial of degree `deg` that is the
+    least squares fit to the data values `y` given at points `x`. If `y` is
+    1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
+    fits are done, one for each column of `y`, and the resulting
+    coefficients are stored in the corresponding columns of a 2-D return.
+    The fitted polynomial(s) are in the form
+
+    .. math::  p(x) = c_0 + c_1 * x + ... + c_n * x^n,
+
+    where `n` is `deg`.
+
+    Parameters
+    ----------
+    x : array_like, shape (`M`,)
+        x-coordinates of the `M` sample (data) points ``(x[i], y[i])``.
+    y : array_like, shape (`M`,) or (`M`, `K`)
+        y-coordinates of the sample points.  Several sets of sample points
+        sharing the same x-coordinates can be (independently) fit with one
+        call to `polyfit` by passing in for `y` a 2-D array that contains
+        one data set per column.
+    deg : int or 1-D array_like
+        Degree(s) of the fitting polynomials. If `deg` is a single integer
+        all terms up to and including the `deg`'th term are included in the
+        fit. For NumPy versions >= 1.11.0 a list of integers specifying the
+        degrees of the terms to include may be used instead.
+    rcond : float, optional
+        Relative condition number of the fit.  Singular values smaller
+        than `rcond`, relative to the largest singular value, will be
+        ignored.  The default value is ``len(x)*eps``, where `eps` is the
+        relative precision of the platform's float type, about 2e-16 in
+        most cases.
+    full : bool, optional
+        Switch determining the nature of the return value.  When ``False``
+        (the default) just the coefficients are returned; when ``True``,
+        diagnostic information from the singular value decomposition (used
+        to solve the fit's matrix equation) is also returned.
+    w : array_like, shape (`M`,), optional
+        Weights. If not None, the weight ``w[i]`` applies to the unsquared
+        residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
+        chosen so that the errors of the products ``w[i]*y[i]`` all have the
+        same variance.  When using inverse-variance weighting, use
+        ``w[i] = 1/sigma(y[i])``.  The default value is None.
+
+    Returns
+    -------
+    coef : ndarray, shape (`deg` + 1,) or (`deg` + 1, `K`)
+        Polynomial coefficients ordered from low to high.  If `y` was 2-D,
+        the coefficients in column `k` of `coef` represent the polynomial
+        fit to the data in `y`'s `k`-th column.
+
+    [residuals, rank, singular_values, rcond] : list
+        These values are only returned if ``full == True``
+
+        - residuals -- sum of squared residuals of the least squares fit
+        - rank -- the numerical rank of the scaled Vandermonde matrix
+        - singular_values -- singular values of the scaled Vandermonde matrix
+        - rcond -- value of `rcond`.
+
+        For more details, see `numpy.linalg.lstsq`.
+
+    Raises
+    ------
+    RankWarning
+        Raised if the matrix in the least-squares fit is rank deficient.
+        The warning is only raised if ``full == False``.  The warnings can
+        be turned off by:
+
+        >>> import warnings
+        >>> warnings.simplefilter('ignore', np.exceptions.RankWarning)
+
+    See Also
+    --------
+    numpy.polynomial.chebyshev.chebfit
+    numpy.polynomial.legendre.legfit
+    numpy.polynomial.laguerre.lagfit
+    numpy.polynomial.hermite.hermfit
+    numpy.polynomial.hermite_e.hermefit
+    polyval : Evaluates a polynomial.
+    polyvander : Vandermonde matrix for powers.
+    numpy.linalg.lstsq : Computes a least-squares fit from the matrix.
+    scipy.interpolate.UnivariateSpline : Computes spline fits.
+
+    Notes
+    -----
+    The solution is the coefficients of the polynomial `p` that minimizes
+    the sum of the weighted squared errors
+
+    .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
+
+    where the :math:`w_j` are the weights. This problem is solved by
+    setting up the (typically) over-determined matrix equation:
+
+    .. math:: V(x) * c = w * y,
+
+    where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the
+    coefficients to be solved for, `w` are the weights, and `y` are the
+    observed values.  This equation is then solved using the singular value
+    decomposition of `V`.
+
+    If some of the singular values of `V` are so small that they are
+    neglected (and `full` == ``False``), a `~exceptions.RankWarning` will be
+    raised.  This means that the coefficient values may be poorly determined.
+    Fitting to a lower order polynomial will usually get rid of the warning
+    (but may not be what you want, of course; if you have independent
+    reason(s) for choosing the degree which isn't working, you may have to:
+    a) reconsider those reasons, and/or b) reconsider the quality of your
+    data).  The `rcond` parameter can also be set to a value smaller than
+    its default, but the resulting fit may be spurious and have large
+    contributions from roundoff error.
+
+    Polynomial fits using double precision tend to "fail" at about
+    (polynomial) degree 20. Fits using Chebyshev or Legendre series are
+    generally better conditioned, but much can still depend on the
+    distribution of the sample points and the smoothness of the data.  If
+    the quality of the fit is inadequate, splines may be a good
+    alternative.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from numpy.polynomial import polynomial as P
+    >>> x = np.linspace(-1,1,51)  # x "data": [-1, -0.96, ..., 0.96, 1]
+    >>> rng = np.random.default_rng()
+    >>> err = rng.normal(size=len(x))
+    >>> y = x**3 - x + err  # x^3 - x + Gaussian noise
+    >>> c, stats = P.polyfit(x,y,3,full=True)
+    >>> c # c[0], c[1] approx. -1, c[2] should be approx. 0, c[3] approx. 1
+    array([ 0.23111996, -1.02785049, -0.2241444 ,  1.08405657]) # may vary
+    >>> stats # note the large SSR, explaining the rather poor results
+    [array([48.312088]),                                        # may vary
+     4,
+     array([1.38446749, 1.32119158, 0.50443316, 0.28853036]),
+     1.1324274851176597e-14]
+
+    Same thing without the added noise
+
+    >>> y = x**3 - x
+    >>> c, stats = P.polyfit(x,y,3,full=True)
+    >>> c # c[0], c[1] ~= -1, c[2] should be "very close to 0", c[3] ~= 1
+    array([-6.73496154e-17, -1.00000000e+00,  0.00000000e+00,  1.00000000e+00])
+    >>> stats # note the minuscule SSR
+    [array([8.79579319e-31]),
+     np.int32(4),
+     array([1.38446749, 1.32119158, 0.50443316, 0.28853036]),
+     1.1324274851176597e-14]
+
+    """
+    return pu._fit(polyvander, x, y, deg, rcond, full, w)
+
+
+def polycompanion(c):
+    """
+    Return the companion matrix of c.
+
+    The companion matrix for power series cannot be made symmetric by
+    scaling the basis, so this function differs from those for the
+    orthogonal polynomials.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of polynomial coefficients ordered from low to high
+        degree.
+
+    Returns
+    -------
+    mat : ndarray
+        Companion matrix of dimensions (deg, deg).
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> c = (1, 2, 3)
+    >>> P.polycompanion(c)
+    array([[ 0.        , -0.33333333],
+           [ 1.        , -0.66666667]])
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) < 2:
+        raise ValueError('Series must have maximum degree of at least 1.')
+    if len(c) == 2:
+        return np.array([[-c[0] / c[1]]])
+
+    n = len(c) - 1
+    mat = np.zeros((n, n), dtype=c.dtype)
+    bot = mat.reshape(-1)[n::n + 1]
+    bot[...] = 1
+    mat[:, -1] -= c[:-1] / c[-1]
+    return mat
+
+
+def polyroots(c):
+    """
+    Compute the roots of a polynomial.
+
+    Return the roots (a.k.a. "zeros") of the polynomial
+
+    .. math:: p(x) = \\sum_i c[i] * x^i.
+
+    Parameters
+    ----------
+    c : 1-D array_like
+        1-D array of polynomial coefficients.
+
+    Returns
+    -------
+    out : ndarray
+        Array of the roots of the polynomial. If all the roots are real,
+        then `out` is also real, otherwise it is complex.
+
+    See Also
+    --------
+    numpy.polynomial.chebyshev.chebroots
+    numpy.polynomial.legendre.legroots
+    numpy.polynomial.laguerre.lagroots
+    numpy.polynomial.hermite.hermroots
+    numpy.polynomial.hermite_e.hermeroots
+
+    Notes
+    -----
+    The root estimates are obtained as the eigenvalues of the companion
+    matrix, Roots far from the origin of the complex plane may have large
+    errors due to the numerical instability of the power series for such
+    values. Roots with multiplicity greater than 1 will also show larger
+    errors as the value of the series near such points is relatively
+    insensitive to errors in the roots. Isolated roots near the origin can
+    be improved by a few iterations of Newton's method.
+
+    Examples
+    --------
+    >>> import numpy.polynomial.polynomial as poly
+    >>> poly.polyroots(poly.polyfromroots((-1,0,1)))
+    array([-1.,  0.,  1.])
+    >>> poly.polyroots(poly.polyfromroots((-1,0,1))).dtype
+    dtype('float64')
+    >>> j = complex(0,1)
+    >>> poly.polyroots(poly.polyfromroots((-j,0,j)))
+    array([  0.00000000e+00+0.j,   0.00000000e+00+1.j,   2.77555756e-17-1.j])  # may vary
+
+    """  # noqa: E501
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) < 2:
+        return np.array([], dtype=c.dtype)
+    if len(c) == 2:
+        return np.array([-c[0] / c[1]])
+
+    m = polycompanion(c)
+    r = np.linalg.eigvals(m)
+    r.sort()
+    return r
+
+
+#
+# polynomial class
+#
+
+class Polynomial(ABCPolyBase):
+    """A power series class.
+
+    The Polynomial class provides the standard Python numerical methods
+    '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the
+    attributes and methods listed below.
+
+    Parameters
+    ----------
+    coef : array_like
+        Polynomial coefficients in order of increasing degree, i.e.,
+        ``(1, 2, 3)`` give ``1 + 2*x + 3*x**2``.
+    domain : (2,) array_like, optional
+        Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
+        to the interval ``[window[0], window[1]]`` by shifting and scaling.
+        The default value is [-1., 1.].
+    window : (2,) array_like, optional
+        Window, see `domain` for its use. The default value is [-1., 1.].
+    symbol : str, optional
+        Symbol used to represent the independent variable in string
+        representations of the polynomial expression, e.g. for printing.
+        The symbol must be a valid Python identifier. Default value is 'x'.
+
+        .. versionadded:: 1.24
+
+    """
+    # Virtual Functions
+    _add = staticmethod(polyadd)
+    _sub = staticmethod(polysub)
+    _mul = staticmethod(polymul)
+    _div = staticmethod(polydiv)
+    _pow = staticmethod(polypow)
+    _val = staticmethod(polyval)
+    _int = staticmethod(polyint)
+    _der = staticmethod(polyder)
+    _fit = staticmethod(polyfit)
+    _line = staticmethod(polyline)
+    _roots = staticmethod(polyroots)
+    _fromroots = staticmethod(polyfromroots)
+
+    # Virtual properties
+    domain = np.array(polydomain)
+    window = np.array(polydomain)
+    basis_name = None
+
+    @classmethod
+    def _str_term_unicode(cls, i, arg_str):
+        if i == '1':
+            return f"·{arg_str}"
+        else:
+            return f"·{arg_str}{i.translate(cls._superscript_mapping)}"
+
+    @staticmethod
+    def _str_term_ascii(i, arg_str):
+        if i == '1':
+            return f" {arg_str}"
+        else:
+            return f" {arg_str}**{i}"
+
+    @staticmethod
+    def _repr_latex_term(i, arg_str, needs_parens):
+        if needs_parens:
+            arg_str = rf"\left({arg_str}\right)"
+        if i == 0:
+            return '1'
+        elif i == 1:
+            return arg_str
+        else:
+            return f"{arg_str}^{{{i}}}"

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy/polynomial/polynomial.pyi

@@ -0,0 +1,109 @@
+from typing import Any, ClassVar, Final, overload
+
+import numpy as np
+import numpy.typing as npt
+from numpy._typing import (
+    _ArrayLikeFloat_co,
+    _ArrayLikeNumber_co,
+    _FloatLike_co,
+    _NumberLike_co,
+)
+
+from ._polybase import ABCPolyBase
+from ._polytypes import (
+    _Array1,
+    _Array2,
+    _ArrayLikeCoef_co,
+    _FuncBinOp,
+    _FuncCompanion,
+    _FuncDer,
+    _FuncFit,
+    _FuncFromRoots,
+    _FuncInteg,
+    _FuncLine,
+    _FuncPow,
+    _FuncRoots,
+    _FuncUnOp,
+    _FuncVal,
+    _FuncVal2D,
+    _FuncVal3D,
+    _FuncVander,
+    _FuncVander2D,
+    _FuncVander3D,
+)
+from .polyutils import trimcoef as polytrim
+
+__all__ = [
+    "polyzero",
+    "polyone",
+    "polyx",
+    "polydomain",
+    "polyline",
+    "polyadd",
+    "polysub",
+    "polymulx",
+    "polymul",
+    "polydiv",
+    "polypow",
+    "polyval",
+    "polyvalfromroots",
+    "polyder",
+    "polyint",
+    "polyfromroots",
+    "polyvander",
+    "polyfit",
+    "polytrim",
+    "polyroots",
+    "Polynomial",
+    "polyval2d",
+    "polyval3d",
+    "polygrid2d",
+    "polygrid3d",
+    "polyvander2d",
+    "polyvander3d",
+    "polycompanion",
+]
+
+polydomain: Final[_Array2[np.float64]] = ...
+polyzero: Final[_Array1[np.int_]] = ...
+polyone: Final[_Array1[np.int_]] = ...
+polyx: Final[_Array2[np.int_]] = ...
+
+polyline: Final[_FuncLine] = ...
+polyfromroots: Final[_FuncFromRoots] = ...
+polyadd: Final[_FuncBinOp] = ...
+polysub: Final[_FuncBinOp] = ...
+polymulx: Final[_FuncUnOp] = ...
+polymul: Final[_FuncBinOp] = ...
+polydiv: Final[_FuncBinOp] = ...
+polypow: Final[_FuncPow] = ...
+polyder: Final[_FuncDer] = ...
+polyint: Final[_FuncInteg] = ...
+polyval: Final[_FuncVal] = ...
+polyval2d: Final[_FuncVal2D] = ...
+polyval3d: Final[_FuncVal3D] = ...
+
+@overload
+def polyvalfromroots(x: _FloatLike_co, r: _FloatLike_co, tensor: bool = True) -> np.float64 | Any: ...
+@overload
+def polyvalfromroots(x: _NumberLike_co, r: _NumberLike_co, tensor: bool = True) -> np.complex128 | Any: ...
+@overload
+def polyvalfromroots(x: _ArrayLikeFloat_co, r: _ArrayLikeFloat_co, tensor: bool = True) -> npt.NDArray[np.float64 | Any]: ...
+@overload
+def polyvalfromroots(x: _ArrayLikeNumber_co, r: _ArrayLikeNumber_co, tensor: bool = True) -> npt.NDArray[np.complex128 | Any]: ...
+@overload
+def polyvalfromroots(x: _ArrayLikeCoef_co, r: _ArrayLikeCoef_co, tensor: bool = True) -> npt.NDArray[np.object_ | Any]: ...
+
+polygrid2d: Final[_FuncVal2D] = ...
+polygrid3d: Final[_FuncVal3D] = ...
+polyvander: Final[_FuncVander] = ...
+polyvander2d: Final[_FuncVander2D] = ...
+polyvander3d: Final[_FuncVander3D] = ...
+polyfit: Final[_FuncFit] = ...
+polycompanion: Final[_FuncCompanion] = ...
+polyroots: Final[_FuncRoots] = ...
+
+class Polynomial(ABCPolyBase[None]):
+    basis_name: ClassVar[None] = None  # pyright: ignore[reportIncompatibleMethodOverride]
+    domain: _Array2[np.float64 | Any] = ...  # pyright: ignore[reportIncompatibleMethodOverride]
+    window: _Array2[np.float64 | Any] = ...  # pyright: ignore[reportIncompatibleMethodOverride]

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy/polynomial/polyutils.py

@@ -0,0 +1,759 @@
+"""
+Utility classes and functions for the polynomial modules.
+
+This module provides: error and warning objects; a polynomial base class;
+and some routines used in both the `polynomial` and `chebyshev` modules.
+
+Functions
+---------
+
+.. autosummary::
+   :toctree: generated/
+
+   as_series    convert list of array_likes into 1-D arrays of common type.
+   trimseq      remove trailing zeros.
+   trimcoef     remove small trailing coefficients.
+   getdomain    return the domain appropriate for a given set of abscissae.
+   mapdomain    maps points between domains.
+   mapparms     parameters of the linear map between domains.
+
+"""
+import functools
+import operator
+import warnings
+
+import numpy as np
+
+__all__ = [
+    'as_series', 'trimseq', 'trimcoef', 'getdomain', 'mapdomain', 'mapparms',
+    'format_float']
+
+#
+# Helper functions to convert inputs to 1-D arrays
+#
+def trimseq(seq):
+    """Remove small Poly series coefficients.
+
+    Parameters
+    ----------
+    seq : sequence
+        Sequence of Poly series coefficients.
+
+    Returns
+    -------
+    series : sequence
+        Subsequence with trailing zeros removed. If the resulting sequence
+        would be empty, return the first element. The returned sequence may
+        or may not be a view.
+
+    Notes
+    -----
+    Do not lose the type info if the sequence contains unknown objects.
+
+    """
+    if len(seq) == 0 or seq[-1] != 0:
+        return seq
+    else:
+        for i in range(len(seq) - 1, -1, -1):
+            if seq[i] != 0:
+                break
+        return seq[:i + 1]
+
+
+def as_series(alist, trim=True):
+    """
+    Return argument as a list of 1-d arrays.
+
+    The returned list contains array(s) of dtype double, complex double, or
+    object.  A 1-d argument of shape ``(N,)`` is parsed into ``N`` arrays of
+    size one; a 2-d argument of shape ``(M,N)`` is parsed into ``M`` arrays
+    of size ``N`` (i.e., is "parsed by row"); and a higher dimensional array
+    raises a Value Error if it is not first reshaped into either a 1-d or 2-d
+    array.
+
+    Parameters
+    ----------
+    alist : array_like
+        A 1- or 2-d array_like
+    trim : boolean, optional
+        When True, trailing zeros are removed from the inputs.
+        When False, the inputs are passed through intact.
+
+    Returns
+    -------
+    [a1, a2,...] : list of 1-D arrays
+        A copy of the input data as a list of 1-d arrays.
+
+    Raises
+    ------
+    ValueError
+        Raised when `as_series` cannot convert its input to 1-d arrays, or at
+        least one of the resulting arrays is empty.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from numpy.polynomial import polyutils as pu
+    >>> a = np.arange(4)
+    >>> pu.as_series(a)
+    [array([0.]), array([1.]), array([2.]), array([3.])]
+    >>> b = np.arange(6).reshape((2,3))
+    >>> pu.as_series(b)
+    [array([0., 1., 2.]), array([3., 4., 5.])]
+
+    >>> pu.as_series((1, np.arange(3), np.arange(2, dtype=np.float16)))
+    [array([1.]), array([0., 1., 2.]), array([0., 1.])]
+
+    >>> pu.as_series([2, [1.1, 0.]])
+    [array([2.]), array([1.1])]
+
+    >>> pu.as_series([2, [1.1, 0.]], trim=False)
+    [array([2.]), array([1.1, 0. ])]
+
+    """
+    arrays = [np.array(a, ndmin=1, copy=None) for a in alist]
+    for a in arrays:
+        if a.size == 0:
+            raise ValueError("Coefficient array is empty")
+        if a.ndim != 1:
+            raise ValueError("Coefficient array is not 1-d")
+    if trim:
+        arrays = [trimseq(a) for a in arrays]
+
+    try:
+        dtype = np.common_type(*arrays)
+    except Exception as e:
+        object_dtype = np.dtypes.ObjectDType()
+        has_one_object_type = False
+        ret = []
+        for a in arrays:
+            if a.dtype != object_dtype:
+                tmp = np.empty(len(a), dtype=object_dtype)
+                tmp[:] = a[:]
+                ret.append(tmp)
+            else:
+                has_one_object_type = True
+                ret.append(a.copy())
+        if not has_one_object_type:
+            raise ValueError("Coefficient arrays have no common type") from e
+    else:
+        ret = [np.array(a, copy=True, dtype=dtype) for a in arrays]
+    return ret
+
+
+def trimcoef(c, tol=0):
+    """
+    Remove "small" "trailing" coefficients from a polynomial.
+
+    "Small" means "small in absolute value" and is controlled by the
+    parameter `tol`; "trailing" means highest order coefficient(s), e.g., in
+    ``[0, 1, 1, 0, 0]`` (which represents ``0 + x + x**2 + 0*x**3 + 0*x**4``)
+    both the 3-rd and 4-th order coefficients would be "trimmed."
+
+    Parameters
+    ----------
+    c : array_like
+        1-d array of coefficients, ordered from lowest order to highest.
+    tol : number, optional
+        Trailing (i.e., highest order) elements with absolute value less
+        than or equal to `tol` (default value is zero) are removed.
+
+    Returns
+    -------
+    trimmed : ndarray
+        1-d array with trailing zeros removed.  If the resulting series
+        would be empty, a series containing a single zero is returned.
+
+    Raises
+    ------
+    ValueError
+        If `tol` < 0
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polyutils as pu
+    >>> pu.trimcoef((0,0,3,0,5,0,0))
+    array([0.,  0.,  3.,  0.,  5.])
+    >>> pu.trimcoef((0,0,1e-3,0,1e-5,0,0),1e-3) # item == tol is trimmed
+    array([0.])
+    >>> i = complex(0,1) # works for complex
+    >>> pu.trimcoef((3e-4,1e-3*(1-i),5e-4,2e-5*(1+i)), 1e-3)
+    array([0.0003+0.j   , 0.001 -0.001j])
+
+    """
+    if tol < 0:
+        raise ValueError("tol must be non-negative")
+
+    [c] = as_series([c])
+    [ind] = np.nonzero(np.abs(c) > tol)
+    if len(ind) == 0:
+        return c[:1] * 0
+    else:
+        return c[:ind[-1] + 1].copy()
+
+def getdomain(x):
+    """
+    Return a domain suitable for given abscissae.
+
+    Find a domain suitable for a polynomial or Chebyshev series
+    defined at the values supplied.
+
+    Parameters
+    ----------
+    x : array_like
+        1-d array of abscissae whose domain will be determined.
+
+    Returns
+    -------
+    domain : ndarray
+        1-d array containing two values.  If the inputs are complex, then
+        the two returned points are the lower left and upper right corners
+        of the smallest rectangle (aligned with the axes) in the complex
+        plane containing the points `x`. If the inputs are real, then the
+        two points are the ends of the smallest interval containing the
+        points `x`.
+
+    See Also
+    --------
+    mapparms, mapdomain
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from numpy.polynomial import polyutils as pu
+    >>> points = np.arange(4)**2 - 5; points
+    array([-5, -4, -1,  4])
+    >>> pu.getdomain(points)
+    array([-5.,  4.])
+    >>> c = np.exp(complex(0,1)*np.pi*np.arange(12)/6) # unit circle
+    >>> pu.getdomain(c)
+    array([-1.-1.j,  1.+1.j])
+
+    """
+    [x] = as_series([x], trim=False)
+    if x.dtype.char in np.typecodes['Complex']:
+        rmin, rmax = x.real.min(), x.real.max()
+        imin, imax = x.imag.min(), x.imag.max()
+        return np.array((complex(rmin, imin), complex(rmax, imax)))
+    else:
+        return np.array((x.min(), x.max()))
+
+def mapparms(old, new):
+    """
+    Linear map parameters between domains.
+
+    Return the parameters of the linear map ``offset + scale*x`` that maps
+    `old` to `new` such that ``old[i] -> new[i]``, ``i = 0, 1``.
+
+    Parameters
+    ----------
+    old, new : array_like
+        Domains. Each domain must (successfully) convert to a 1-d array
+        containing precisely two values.
+
+    Returns
+    -------
+    offset, scale : scalars
+        The map ``L(x) = offset + scale*x`` maps the first domain to the
+        second.
+
+    See Also
+    --------
+    getdomain, mapdomain
+
+    Notes
+    -----
+    Also works for complex numbers, and thus can be used to calculate the
+    parameters required to map any line in the complex plane to any other
+    line therein.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polyutils as pu
+    >>> pu.mapparms((-1,1),(-1,1))
+    (0.0, 1.0)
+    >>> pu.mapparms((1,-1),(-1,1))
+    (-0.0, -1.0)
+    >>> i = complex(0,1)
+    >>> pu.mapparms((-i,-1),(1,i))
+    ((1+1j), (1-0j))
+
+    """
+    oldlen = old[1] - old[0]
+    newlen = new[1] - new[0]
+    off = (old[1] * new[0] - old[0] * new[1]) / oldlen
+    scl = newlen / oldlen
+    return off, scl
+
+def mapdomain(x, old, new):
+    """
+    Apply linear map to input points.
+
+    The linear map ``offset + scale*x`` that maps the domain `old` to
+    the domain `new` is applied to the points `x`.
+
+    Parameters
+    ----------
+    x : array_like
+        Points to be mapped. If `x` is a subtype of ndarray the subtype
+        will be preserved.
+    old, new : array_like
+        The two domains that determine the map.  Each must (successfully)
+        convert to 1-d arrays containing precisely two values.
+
+    Returns
+    -------
+    x_out : ndarray
+        Array of points of the same shape as `x`, after application of the
+        linear map between the two domains.
+
+    See Also
+    --------
+    getdomain, mapparms
+
+    Notes
+    -----
+    Effectively, this implements:
+
+    .. math::
+        x\\_out = new[0] + m(x - old[0])
+
+    where
+
+    .. math::
+        m = \\frac{new[1]-new[0]}{old[1]-old[0]}
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from numpy.polynomial import polyutils as pu
+    >>> old_domain = (-1,1)
+    >>> new_domain = (0,2*np.pi)
+    >>> x = np.linspace(-1,1,6); x
+    array([-1. , -0.6, -0.2,  0.2,  0.6,  1. ])
+    >>> x_out = pu.mapdomain(x, old_domain, new_domain); x_out
+    array([ 0.        ,  1.25663706,  2.51327412,  3.76991118,  5.02654825, # may vary
+            6.28318531])
+    >>> x - pu.mapdomain(x_out, new_domain, old_domain)
+    array([0., 0., 0., 0., 0., 0.])
+
+    Also works for complex numbers (and thus can be used to map any line in
+    the complex plane to any other line therein).
+
+    >>> i = complex(0,1)
+    >>> old = (-1 - i, 1 + i)
+    >>> new = (-1 + i, 1 - i)
+    >>> z = np.linspace(old[0], old[1], 6); z
+    array([-1. -1.j , -0.6-0.6j, -0.2-0.2j,  0.2+0.2j,  0.6+0.6j,  1. +1.j ])
+    >>> new_z = pu.mapdomain(z, old, new); new_z
+    array([-1.0+1.j , -0.6+0.6j, -0.2+0.2j,  0.2-0.2j,  0.6-0.6j,  1.0-1.j ]) # may vary
+
+    """
+    if type(x) not in (int, float, complex) and not isinstance(x, np.generic):
+        x = np.asanyarray(x)
+    off, scl = mapparms(old, new)
+    return off + scl * x
+
+
+def _nth_slice(i, ndim):
+    sl = [np.newaxis] * ndim
+    sl[i] = slice(None)
+    return tuple(sl)
+
+
+def _vander_nd(vander_fs, points, degrees):
+    r"""
+    A generalization of the Vandermonde matrix for N dimensions
+
+    The result is built by combining the results of 1d Vandermonde matrices,
+
+    .. math::
+        W[i_0, \ldots, i_M, j_0, \ldots, j_N] = \prod_{k=0}^N{V_k(x_k)[i_0, \ldots, i_M, j_k]}
+
+    where
+
+    .. math::
+        N &= \texttt{len(points)} = \texttt{len(degrees)} = \texttt{len(vander\_fs)} \\
+        M &= \texttt{points[k].ndim} \\
+        V_k &= \texttt{vander\_fs[k]} \\
+        x_k &= \texttt{points[k]} \\
+        0 \le j_k &\le \texttt{degrees[k]}
+
+    Expanding the one-dimensional :math:`V_k` functions gives:
+
+    .. math::
+        W[i_0, \ldots, i_M, j_0, \ldots, j_N] = \prod_{k=0}^N{B_{k, j_k}(x_k[i_0, \ldots, i_M])}
+
+    where :math:`B_{k,m}` is the m'th basis of the polynomial construction used along
+    dimension :math:`k`. For a regular polynomial, :math:`B_{k, m}(x) = P_m(x) = x^m`.
+
+    Parameters
+    ----------
+    vander_fs : Sequence[function(array_like, int) -> ndarray]
+        The 1d vander function to use for each axis, such as ``polyvander``
+    points : Sequence[array_like]
+        Arrays of point coordinates, all of the same shape. The dtypes
+        will be converted to either float64 or complex128 depending on
+        whether any of the elements are complex. Scalars are converted to
+        1-D arrays.
+        This must be the same length as `vander_fs`.
+    degrees : Sequence[int]
+        The maximum degree (inclusive) to use for each axis.
+        This must be the same length as `vander_fs`.
+
+    Returns
+    -------
+    vander_nd : ndarray
+        An array of shape ``points[0].shape + tuple(d + 1 for d in degrees)``.
+    """  # noqa: E501
+    n_dims = len(vander_fs)
+    if n_dims != len(points):
+        raise ValueError(
+            f"Expected {n_dims} dimensions of sample points, got {len(points)}")
+    if n_dims != len(degrees):
+        raise ValueError(
+            f"Expected {n_dims} dimensions of degrees, got {len(degrees)}")
+    if n_dims == 0:
+        raise ValueError("Unable to guess a dtype or shape when no points are given")
+
+    # convert to the same shape and type
+    points = tuple(np.asarray(tuple(points)) + 0.0)
+
+    # produce the vandermonde matrix for each dimension, placing the last
+    # axis of each in an independent trailing axis of the output
+    vander_arrays = (
+        vander_fs[i](points[i], degrees[i])[(...,) + _nth_slice(i, n_dims)]
+        for i in range(n_dims)
+    )
+
+    # we checked this wasn't empty already, so no `initial` needed
+    return functools.reduce(operator.mul, vander_arrays)
+
+
+def _vander_nd_flat(vander_fs, points, degrees):
+    """
+    Like `_vander_nd`, but flattens the last ``len(degrees)`` axes into a single axis
+
+    Used to implement the public ``<type>vander<n>d`` functions.
+    """
+    v = _vander_nd(vander_fs, points, degrees)
+    return v.reshape(v.shape[:-len(degrees)] + (-1,))
+
+
+def _fromroots(line_f, mul_f, roots):
+    """
+    Helper function used to implement the ``<type>fromroots`` functions.
+
+    Parameters
+    ----------
+    line_f : function(float, float) -> ndarray
+        The ``<type>line`` function, such as ``polyline``
+    mul_f : function(array_like, array_like) -> ndarray
+        The ``<type>mul`` function, such as ``polymul``
+    roots
+        See the ``<type>fromroots`` functions for more detail
+    """
+    if len(roots) == 0:
+        return np.ones(1)
+    else:
+        [roots] = as_series([roots], trim=False)
+        roots.sort()
+        p = [line_f(-r, 1) for r in roots]
+        n = len(p)
+        while n > 1:
+            m, r = divmod(n, 2)
+            tmp = [mul_f(p[i], p[i + m]) for i in range(m)]
+            if r:
+                tmp[0] = mul_f(tmp[0], p[-1])
+            p = tmp
+            n = m
+        return p[0]
+
+
+def _valnd(val_f, c, *args):
+    """
+    Helper function used to implement the ``<type>val<n>d`` functions.
+
+    Parameters
+    ----------
+    val_f : function(array_like, array_like, tensor: bool) -> array_like
+        The ``<type>val`` function, such as ``polyval``
+    c, args
+        See the ``<type>val<n>d`` functions for more detail
+    """
+    args = [np.asanyarray(a) for a in args]
+    shape0 = args[0].shape
+    if not all(a.shape == shape0 for a in args[1:]):
+        if len(args) == 3:
+            raise ValueError('x, y, z are incompatible')
+        elif len(args) == 2:
+            raise ValueError('x, y are incompatible')
+        else:
+            raise ValueError('ordinates are incompatible')
+    it = iter(args)
+    x0 = next(it)
+
+    # use tensor on only the first
+    c = val_f(x0, c)
+    for xi in it:
+        c = val_f(xi, c, tensor=False)
+    return c
+
+
+def _gridnd(val_f, c, *args):
+    """
+    Helper function used to implement the ``<type>grid<n>d`` functions.
+
+    Parameters
+    ----------
+    val_f : function(array_like, array_like, tensor: bool) -> array_like
+        The ``<type>val`` function, such as ``polyval``
+    c, args
+        See the ``<type>grid<n>d`` functions for more detail
+    """
+    for xi in args:
+        c = val_f(xi, c)
+    return c
+
+
+def _div(mul_f, c1, c2):
+    """
+    Helper function used to implement the ``<type>div`` functions.
+
+    Implementation uses repeated subtraction of c2 multiplied by the nth basis.
+    For some polynomial types, a more efficient approach may be possible.
+
+    Parameters
+    ----------
+    mul_f : function(array_like, array_like) -> array_like
+        The ``<type>mul`` function, such as ``polymul``
+    c1, c2
+        See the ``<type>div`` functions for more detail
+    """
+    # c1, c2 are trimmed copies
+    [c1, c2] = as_series([c1, c2])
+    if c2[-1] == 0:
+        raise ZeroDivisionError  # FIXME: add message with details to exception
+
+    lc1 = len(c1)
+    lc2 = len(c2)
+    if lc1 < lc2:
+        return c1[:1] * 0, c1
+    elif lc2 == 1:
+        return c1 / c2[-1], c1[:1] * 0
+    else:
+        quo = np.empty(lc1 - lc2 + 1, dtype=c1.dtype)
+        rem = c1
+        for i in range(lc1 - lc2, - 1, -1):
+            p = mul_f([0] * i + [1], c2)
+            q = rem[-1] / p[-1]
+            rem = rem[:-1] - q * p[:-1]
+            quo[i] = q
+        return quo, trimseq(rem)
+
+
+def _add(c1, c2):
+    """ Helper function used to implement the ``<type>add`` functions. """
+    # c1, c2 are trimmed copies
+    [c1, c2] = as_series([c1, c2])
+    if len(c1) > len(c2):
+        c1[:c2.size] += c2
+        ret = c1
+    else:
+        c2[:c1.size] += c1
+        ret = c2
+    return trimseq(ret)
+
+
+def _sub(c1, c2):
+    """ Helper function used to implement the ``<type>sub`` functions. """
+    # c1, c2 are trimmed copies
+    [c1, c2] = as_series([c1, c2])
+    if len(c1) > len(c2):
+        c1[:c2.size] -= c2
+        ret = c1
+    else:
+        c2 = -c2
+        c2[:c1.size] += c1
+        ret = c2
+    return trimseq(ret)
+
+
+def _fit(vander_f, x, y, deg, rcond=None, full=False, w=None):
+    """
+    Helper function used to implement the ``<type>fit`` functions.
+
+    Parameters
+    ----------
+    vander_f : function(array_like, int) -> ndarray
+        The 1d vander function, such as ``polyvander``
+    c1, c2
+        See the ``<type>fit`` functions for more detail
+    """
+    x = np.asarray(x) + 0.0
+    y = np.asarray(y) + 0.0
+    deg = np.asarray(deg)
+
+    # check arguments.
+    if deg.ndim > 1 or deg.dtype.kind not in 'iu' or deg.size == 0:
+        raise TypeError("deg must be an int or non-empty 1-D array of int")
+    if deg.min() < 0:
+        raise ValueError("expected deg >= 0")
+    if x.ndim != 1:
+        raise TypeError("expected 1D vector for x")
+    if x.size == 0:
+        raise TypeError("expected non-empty vector for x")
+    if y.ndim < 1 or y.ndim > 2:
+        raise TypeError("expected 1D or 2D array for y")
+    if len(x) != len(y):
+        raise TypeError("expected x and y to have same length")
+
+    if deg.ndim == 0:
+        lmax = deg
+        order = lmax + 1
+        van = vander_f(x, lmax)
+    else:
+        deg = np.sort(deg)
+        lmax = deg[-1]
+        order = len(deg)
+        van = vander_f(x, lmax)[:, deg]
+
+    # set up the least squares matrices in transposed form
+    lhs = van.T
+    rhs = y.T
+    if w is not None:
+        w = np.asarray(w) + 0.0
+        if w.ndim != 1:
+            raise TypeError("expected 1D vector for w")
+        if len(x) != len(w):
+            raise TypeError("expected x and w to have same length")
+        # apply weights. Don't use inplace operations as they
+        # can cause problems with NA.
+        lhs = lhs * w
+        rhs = rhs * w
+
+    # set rcond
+    if rcond is None:
+        rcond = len(x) * np.finfo(x.dtype).eps
+
+    # Determine the norms of the design matrix columns.
+    if issubclass(lhs.dtype.type, np.complexfloating):
+        scl = np.sqrt((np.square(lhs.real) + np.square(lhs.imag)).sum(1))
+    else:
+        scl = np.sqrt(np.square(lhs).sum(1))
+    scl[scl == 0] = 1
+
+    # Solve the least squares problem.
+    c, resids, rank, s = np.linalg.lstsq(lhs.T / scl, rhs.T, rcond)
+    c = (c.T / scl).T
+
+    # Expand c to include non-fitted coefficients which are set to zero
+    if deg.ndim > 0:
+        if c.ndim == 2:
+            cc = np.zeros((lmax + 1, c.shape[1]), dtype=c.dtype)
+        else:
+            cc = np.zeros(lmax + 1, dtype=c.dtype)
+        cc[deg] = c
+        c = cc
+
+    # warn on rank reduction
+    if rank != order and not full:
+        msg = "The fit may be poorly conditioned"
+        warnings.warn(msg, np.exceptions.RankWarning, stacklevel=2)
+
+    if full:
+        return c, [resids, rank, s, rcond]
+    else:
+        return c
+
+
+def _pow(mul_f, c, pow, maxpower):
+    """
+    Helper function used to implement the ``<type>pow`` functions.
+
+    Parameters
+    ----------
+    mul_f : function(array_like, array_like) -> ndarray
+        The ``<type>mul`` function, such as ``polymul``
+    c : array_like
+        1-D array of array of series coefficients
+    pow, maxpower
+        See the ``<type>pow`` functions for more detail
+    """
+    # c is a trimmed copy
+    [c] = as_series([c])
+    power = int(pow)
+    if power != pow or power < 0:
+        raise ValueError("Power must be a non-negative integer.")
+    elif maxpower is not None and power > maxpower:
+        raise ValueError("Power is too large")
+    elif power == 0:
+        return np.array([1], dtype=c.dtype)
+    elif power == 1:
+        return c
+    else:
+        # This can be made more efficient by using powers of two
+        # in the usual way.
+        prd = c
+        for i in range(2, power + 1):
+            prd = mul_f(prd, c)
+        return prd
+
+
+def _as_int(x, desc):
+    """
+    Like `operator.index`, but emits a custom exception when passed an
+    incorrect type
+
+    Parameters
+    ----------
+    x : int-like
+        Value to interpret as an integer
+    desc : str
+        description to include in any error message
+
+    Raises
+    ------
+    TypeError : if x is a float or non-numeric
+    """
+    try:
+        return operator.index(x)
+    except TypeError as e:
+        raise TypeError(f"{desc} must be an integer, received {x}") from e
+
+
+def format_float(x, parens=False):
+    from numpy._core.multiarray import dragon4_positional, dragon4_scientific
+
+    if not np.issubdtype(type(x), np.floating):
+        return str(x)
+
+    opts = np.get_printoptions()
+
+    if np.isnan(x):
+        return opts['nanstr']
+    elif np.isinf(x):
+        return opts['infstr']
+
+    exp_format = False
+    if x != 0:
+        a = np.abs(x)
+        if a >= 1.e8 or a < 10**min(0, -(opts['precision'] - 1) // 2):
+            exp_format = True
+
+    trim, unique = '0', True
+    if opts['floatmode'] == 'fixed':
+        trim, unique = 'k', False
+
+    if exp_format:
+        s = dragon4_scientific(x, precision=opts['precision'],
+                               unique=unique, trim=trim,
+                               sign=opts['sign'] == '+')
+        if parens:
+            s = '(' + s + ')'
+    else:
+        s = dragon4_positional(x, precision=opts['precision'],
+                               fractional=True,
+                               unique=unique, trim=trim,
+                               sign=opts['sign'] == '+')
+    return s

physworld_p1_20261018/envs/p1/lib/python3.11/site-packages/numpy/polynomial/polyutils.pyi

@@ -0,0 +1,307 @@
+from collections.abc import Callable, Iterable, Sequence
+from typing import (
+    Final,
+    Literal,
+    Protocol,
+    SupportsIndex,
+    TypeAlias,
+    TypeVar,
+    overload,
+    type_check_only,
+)
+
+import numpy as np
+import numpy.typing as npt
+from numpy._typing import (
+    _ArrayLikeComplex_co,
+    _ArrayLikeFloat_co,
+    _ArrayLikeObject_co,
+    _FloatLike_co,
+    _NumberLike_co,
+)
+
+from ._polytypes import (
+    _AnyInt,
+    _Array2,
+    _ArrayLikeCoef_co,
+    _CoefArray,
+    _CoefLike_co,
+    _CoefSeries,
+    _ComplexArray,
+    _ComplexSeries,
+    _FloatArray,
+    _FloatSeries,
+    _FuncBinOp,
+    _ObjectArray,
+    _ObjectSeries,
+    _SeriesLikeCoef_co,
+    _SeriesLikeComplex_co,
+    _SeriesLikeFloat_co,
+    _SeriesLikeInt_co,
+    _SeriesLikeObject_co,
+    _Tuple2,
+)
+
+__all__ = ["as_series", "format_float", "getdomain", "mapdomain", "mapparms", "trimcoef", "trimseq"]
+
+_T = TypeVar("_T")
+_SeqT = TypeVar("_SeqT", bound=_CoefArray | Sequence[_CoefLike_co])
+
+_AnyLineF: TypeAlias = Callable[[float, float], _CoefArray]
+_AnyMulF: TypeAlias = Callable[[np.ndarray | list[int], np.ndarray], _CoefArray]
+_AnyVanderF: TypeAlias = Callable[[np.ndarray, int], _CoefArray]
+
+@type_check_only
+class _ValFunc(Protocol[_T]):
+    def __call__(self, x: np.ndarray, c: _T, /, *, tensor: bool = True) -> _T: ...
+
+###
+
+@overload
+def as_series(alist: npt.NDArray[np.integer] | _FloatArray, trim: bool = True) -> list[_FloatSeries]: ...
+@overload
+def as_series(alist: _ComplexArray, trim: bool = True) -> list[_ComplexSeries]: ...
+@overload
+def as_series(alist: _ObjectArray, trim: bool = True) -> list[_ObjectSeries]: ...
+@overload
+def as_series(alist: Iterable[_FloatArray | npt.NDArray[np.integer]], trim: bool = True) -> list[_FloatSeries]: ...
+@overload
+def as_series(alist: Iterable[_ComplexArray], trim: bool = True) -> list[_ComplexSeries]: ...
+@overload
+def as_series(alist: Iterable[_ObjectArray], trim: bool = True) -> list[_ObjectSeries]: ...
+@overload
+def as_series(alist: Iterable[_SeriesLikeFloat_co | float], trim: bool = True) -> list[_FloatSeries]: ...
+@overload
+def as_series(alist: Iterable[_SeriesLikeComplex_co | complex], trim: bool = True) -> list[_ComplexSeries]: ...
+@overload
+def as_series(alist: Iterable[_SeriesLikeCoef_co | object], trim: bool = True) -> list[_ObjectSeries]: ...
+
+#
+def trimseq(seq: _SeqT) -> _SeqT: ...
+
+#
+@overload
+def trimcoef(c: npt.NDArray[np.integer] | _FloatArray, tol: _FloatLike_co = 0) -> _FloatSeries: ...
+@overload
+def trimcoef(c: _ComplexArray, tol: _FloatLike_co = 0) -> _ComplexSeries: ...
+@overload
+def trimcoef(c: _ObjectArray, tol: _FloatLike_co = 0) -> _ObjectSeries: ...
+@overload
+def trimcoef(c: _SeriesLikeFloat_co | float, tol: _FloatLike_co = 0) -> _FloatSeries: ...
+@overload
+def trimcoef(c: _SeriesLikeComplex_co | complex, tol: _FloatLike_co = 0) -> _ComplexSeries: ...
+@overload
+def trimcoef(c: _SeriesLikeCoef_co | object, tol: _FloatLike_co = 0) -> _ObjectSeries: ...
+
+#
+@overload
+def getdomain(x: _FloatArray | npt.NDArray[np.integer]) -> _Array2[np.float64]: ...
+@overload
+def getdomain(x: _ComplexArray) -> _Array2[np.complex128]: ...
+@overload
+def getdomain(x: _ObjectArray) -> _Array2[np.object_]: ...
+@overload
+def getdomain(x: _SeriesLikeFloat_co | float) -> _Array2[np.float64]: ...
+@overload
+def getdomain(x: _SeriesLikeComplex_co | complex) -> _Array2[np.complex128]: ...
+@overload
+def getdomain(x: _SeriesLikeCoef_co | object) -> _Array2[np.object_]: ...
+
+#
+@overload
+def mapparms(old: npt.NDArray[np.floating | np.integer], new: npt.NDArray[np.floating | np.integer]) -> _Tuple2[np.floating]: ...
+@overload
+def mapparms(old: npt.NDArray[np.number], new: npt.NDArray[np.number]) -> _Tuple2[np.complexfloating]: ...
+@overload
+def mapparms(old: npt.NDArray[np.object_ | np.number], new: npt.NDArray[np.object_ | np.number]) -> _Tuple2[object]: ...
+@overload
+def mapparms(old: Sequence[float], new: Sequence[float]) -> _Tuple2[float]: ...
+@overload
+def mapparms(old: Sequence[complex], new: Sequence[complex]) -> _Tuple2[complex]: ...
+@overload
+def mapparms(old: _SeriesLikeFloat_co, new: _SeriesLikeFloat_co) -> _Tuple2[np.floating]: ...
+@overload
+def mapparms(old: _SeriesLikeComplex_co, new: _SeriesLikeComplex_co) -> _Tuple2[np.complexfloating]: ...
+@overload
+def mapparms(old: _SeriesLikeCoef_co, new: _SeriesLikeCoef_co) -> _Tuple2[object]: ...
+
+#
+@overload
+def mapdomain(x: _FloatLike_co, old: _SeriesLikeFloat_co, new: _SeriesLikeFloat_co) -> np.floating: ...
+@overload
+def mapdomain(x: _NumberLike_co, old: _SeriesLikeComplex_co, new: _SeriesLikeComplex_co) -> np.complexfloating: ...
+@overload
+def mapdomain(
+    x: npt.NDArray[np.floating | np.integer],
+    old: npt.NDArray[np.floating | np.integer],
+    new: npt.NDArray[np.floating | np.integer],
+) -> _FloatSeries: ...
+@overload
+def mapdomain(x: npt.NDArray[np.number], old: npt.NDArray[np.number], new: npt.NDArray[np.number]) -> _ComplexSeries: ...
+@overload
+def mapdomain(
+    x: npt.NDArray[np.object_ | np.number],
+    old: npt.NDArray[np.object_ | np.number],
+    new: npt.NDArray[np.object_ | np.number],
+) -> _ObjectSeries: ...
+@overload
+def mapdomain(x: _SeriesLikeFloat_co, old: _SeriesLikeFloat_co, new: _SeriesLikeFloat_co) -> _FloatSeries: ...
+@overload
+def mapdomain(x: _SeriesLikeComplex_co, old: _SeriesLikeComplex_co, new: _SeriesLikeComplex_co) -> _ComplexSeries: ...
+@overload
+def mapdomain(x: _SeriesLikeCoef_co, old: _SeriesLikeCoef_co, new: _SeriesLikeCoef_co) -> _ObjectSeries: ...
+@overload
+def mapdomain(x: _CoefLike_co, old: _SeriesLikeCoef_co, new: _SeriesLikeCoef_co) -> object: ...
+
+#
+def _nth_slice(i: SupportsIndex, ndim: SupportsIndex) -> tuple[slice | None, ...]: ...
+
+# keep in sync with `vander_nd_flat`
+@overload
+def _vander_nd(
+    vander_fs: Sequence[_AnyVanderF],
+    points: Sequence[_ArrayLikeFloat_co],
+    degrees: Sequence[SupportsIndex],
+) -> _FloatArray: ...
+@overload
+def _vander_nd(
+    vander_fs: Sequence[_AnyVanderF],
+    points: Sequence[_ArrayLikeComplex_co],
+    degrees: Sequence[SupportsIndex],
+) -> _ComplexArray: ...
+@overload
+def _vander_nd(
+    vander_fs: Sequence[_AnyVanderF],
+    points: Sequence[_ArrayLikeObject_co | _ArrayLikeComplex_co],
+    degrees: Sequence[SupportsIndex],
+) -> _ObjectArray: ...
+@overload
+def _vander_nd(
+    vander_fs: Sequence[_AnyVanderF],
+    points: Sequence[npt.ArrayLike],
+    degrees: Sequence[SupportsIndex],
+) -> _CoefArray: ...
+
+# keep in sync with `vander_nd`
+@overload
+def _vander_nd_flat(
+    vander_fs: Sequence[_AnyVanderF],
+    points: Sequence[_ArrayLikeFloat_co],
+    degrees: Sequence[SupportsIndex],
+) -> _FloatArray: ...
+@overload
+def _vander_nd_flat(
+    vander_fs: Sequence[_AnyVanderF],
+    points: Sequence[_ArrayLikeComplex_co],
+    degrees: Sequence[SupportsIndex],
+) -> _ComplexArray: ...
+@overload
+def _vander_nd_flat(
+    vander_fs: Sequence[_AnyVanderF],
+    points: Sequence[_ArrayLikeObject_co | _ArrayLikeComplex_co],
+    degrees: Sequence[SupportsIndex],
+) -> _ObjectArray: ...
+@overload
+def _vander_nd_flat(
+    vander_fs: Sequence[_AnyVanderF],
+    points: Sequence[npt.ArrayLike],
+    degrees: Sequence[SupportsIndex],
+) -> _CoefArray: ...
+
+# keep in sync with `._polytypes._FuncFromRoots`
+@overload
+def _fromroots(line_f: _AnyLineF, mul_f: _AnyMulF, roots: _SeriesLikeFloat_co) -> _FloatSeries: ...
+@overload
+def _fromroots(line_f: _AnyLineF, mul_f: _AnyMulF, roots: _SeriesLikeComplex_co) -> _ComplexSeries: ...
+@overload
+def _fromroots(line_f: _AnyLineF, mul_f: _AnyMulF, roots: _SeriesLikeObject_co) -> _ObjectSeries: ...
+@overload
+def _fromroots(line_f: _AnyLineF, mul_f: _AnyMulF, roots: _SeriesLikeCoef_co) -> _CoefSeries: ...
+
+# keep in sync with `_gridnd`
+def _valnd(val_f: _ValFunc[_T], c: _T, *args: npt.ArrayLike) -> _T: ...
+
+# keep in sync with `_valnd`
+def _gridnd(val_f: _ValFunc[_T], c: _T, *args: npt.ArrayLike) -> _T: ...
+
+# keep in sync with `_polytypes._FuncBinOp`
+@overload
+def _div(mul_f: _AnyMulF, c1: _SeriesLikeFloat_co, c2: _SeriesLikeFloat_co) -> _Tuple2[_FloatSeries]: ...
+@overload
+def _div(mul_f: _AnyMulF, c1: _SeriesLikeComplex_co, c2: _SeriesLikeComplex_co) -> _Tuple2[_ComplexSeries]: ...
+@overload
+def _div(mul_f: _AnyMulF, c1: _SeriesLikeObject_co, c2: _SeriesLikeObject_co) -> _Tuple2[_ObjectSeries]: ...
+@overload
+def _div(mul_f: _AnyMulF, c1: _SeriesLikeCoef_co, c2: _SeriesLikeCoef_co) -> _Tuple2[_CoefSeries]: ...
+
+_add: Final[_FuncBinOp] = ...
+_sub: Final[_FuncBinOp] = ...
+
+# keep in sync with `_polytypes._FuncPow`
+@overload
+def _pow(mul_f: _AnyMulF, c: _SeriesLikeFloat_co, pow: _AnyInt, maxpower: _AnyInt | None) -> _FloatSeries: ...
+@overload
+def _pow(mul_f: _AnyMulF, c: _SeriesLikeComplex_co, pow: _AnyInt, maxpower: _AnyInt | None) -> _ComplexSeries: ...
+@overload
+def _pow(mul_f: _AnyMulF, c: _SeriesLikeObject_co, pow: _AnyInt, maxpower: _AnyInt | None) -> _ObjectSeries: ...
+@overload
+def _pow(mul_f: _AnyMulF, c: _SeriesLikeCoef_co, pow: _AnyInt, maxpower: _AnyInt | None) -> _CoefSeries: ...
+
+# keep in sync with `_polytypes._FuncFit`
+@overload
+def _fit(
+    vander_f: _AnyVanderF,
+    x: _SeriesLikeFloat_co,
+    y: _ArrayLikeFloat_co,
+    deg: _SeriesLikeInt_co,
+    rcond: _FloatLike_co | None = None,
+    full: Literal[False] = False,
+    w: _SeriesLikeFloat_co | None = None,
+) -> _FloatArray: ...
+@overload
+def _fit(
+    vander_f: _AnyVanderF,
+    x: _SeriesLikeComplex_co,
+    y: _ArrayLikeComplex_co,
+    deg: _SeriesLikeInt_co,
+    rcond: _FloatLike_co | None = None,
+    full: Literal[False] = False,
+    w: _SeriesLikeComplex_co | None = None,
+) -> _ComplexArray: ...
+@overload
+def _fit(
+    vander_f: _AnyVanderF,
+    x: _SeriesLikeCoef_co,
+    y: _ArrayLikeCoef_co,
+    deg: _SeriesLikeInt_co,
+    rcond: _FloatLike_co | None = None,
+    full: Literal[False] = False,
+    w: _SeriesLikeCoef_co | None = None,
+) -> _CoefArray: ...
+@overload
+def _fit(
+    vander_f: _AnyVanderF,
+    x: _SeriesLikeCoef_co,
+    y: _SeriesLikeCoef_co,
+    deg: _SeriesLikeInt_co,
+    rcond: _FloatLike_co | None,
+    full: Literal[True],
+    w: _SeriesLikeCoef_co | None = None,
+) -> tuple[_CoefSeries, Sequence[np.inexact | np.int32]]: ...
+@overload
+def _fit(
+    vander_f: _AnyVanderF,
+    x: _SeriesLikeCoef_co,
+    y: _SeriesLikeCoef_co,
+    deg: _SeriesLikeInt_co,
+    rcond: _FloatLike_co | None = None,
+    *,
+    full: Literal[True],
+    w: _SeriesLikeCoef_co | None = None,
+) -> tuple[_CoefSeries, Sequence[np.inexact | np.int32]]: ...
+
+#
+def _as_int(x: SupportsIndex, desc: str) -> int: ...
+
+#
+def format_float(x: _FloatLike_co, parens: bool = False) -> str: ...