| """Quantitative extremal-eigenspace inheritance and theta examples.""" |
| from __future__ import annotations |
| import numpy as np |
| from scipy.linalg import eigh |
|
|
|
|
| def inherited_gap_bound(lam: float, gap: float, residue: np.ndarray, path_gram: np.ndarray) -> dict: |
| if not np.isfinite(lam) or not np.isfinite(gap) or lam < 0 or gap <= 0: |
| raise ValueError('A nonnegative threshold and positive next gap are required.') |
| R=np.asarray(residue,dtype=float); S=np.asarray(path_gram,dtype=float) |
| if R.ndim != 2 or R.shape[0] != R.shape[1] or S.shape != R.shape: |
| raise ValueError('Compatible square residue and path-Gram matrices are required.') |
| if not np.all(np.isfinite(R)) or not np.all(np.isfinite(S)): |
| raise ValueError('Matrices must have finite entries.') |
| if R.shape == (0,0): |
| return dict(rho=0.0, gap_lower_bound=0.0) |
| if not np.allclose(R,R.T) or not np.allclose(S,S.T): |
| raise ValueError('Matrices must be symmetric.') |
| if np.linalg.eigvalsh(S)[0] <= 0: |
| raise ValueError('The path Gram matrix must be positive definite.') |
| if np.linalg.eigvalsh(R)[0] < -1e-10: |
| raise ValueError('The residue must be positive semidefinite.') |
| rho=max(0.0,float(eigh(R,S,eigvals_only=True)[-1])) |
| return dict(rho=rho, gap_lower_bound=gap*rho/(lam+gap+rho)) |
|
|
|
|
| def theta_bound(lengths, k: int) -> dict: |
| a,b,c=map(float,lengths) |
| if not all(np.isfinite(x) for x in (a,b,c)) or min(a,b,c)<=0 or not isinstance(k,int) or k<2: |
| raise ValueError('Positive lengths and k>=2 are required.') |
| L=a+b+c; mu=np.pi*(k-1)/L; lam=mu*mu |
| |
| mismatch=np.sqrt(2/L)*np.array([1-np.cos(mu*(a+b)), np.cos(mu*L)-np.cos(mu*b)]) |
| R=np.outer(mismatch,mismatch) |
| S=np.array([[a+b,-a],[-a,a+c]]) |
| g0=np.pi**2*(2*k-1)/L**2 |
| result=inherited_gap_bound(lam,g0,R,S) |
| result.update(threshold=lam, next_tree_gap=g0, residue=R.tolist(), path_gram=S.tolist(), k=k, lengths=[a,b,c]) |
| return result |
|
|
