meanas/fdfd_tools/eigensolvers.py

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"""
Solvers for eigenvalue / eigenvector problems
"""
from typing import Tuple, List
import numpy
from numpy.linalg import norm
from scipy import sparse
import scipy.sparse.linalg as spalg
def power_iteration(operator: sparse.spmatrix,
guess_vector: numpy.ndarray = None,
iterations: int = 20,
) -> Tuple[complex, numpy.ndarray]:
"""
Use power iteration to estimate the dominant eigenvector of a matrix.
:param operator: Matrix to analyze.
:param guess_vector: Starting point for the eigenvector. Default is a randomly chosen vector.
:param iterations: Number of iterations to perform. Default 20.
:return: (Largest-magnitude eigenvalue, Corresponding eigenvector estimate)
"""
if numpy.any(numpy.equal(guess_vector, None)):
v = numpy.random.rand(operator.shape[0])
else:
v = guess_vector
for _ in range(iterations):
v = operator @ v
v /= norm(v)
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lm_eigval = v.conj() @ (operator @ v)
return lm_eigval, v
def rayleigh_quotient_iteration(operator: sparse.spmatrix or spalg.LinearOperator,
guess_vector: numpy.ndarray,
iterations: int = 40,
tolerance: float = 1e-13,
solver=None,
) -> Tuple[complex, numpy.ndarray]:
"""
Use Rayleigh quotient iteration to refine an eigenvector guess.
:param operator: Matrix to analyze.
:param guess_vector: Eigenvector to refine.
:param iterations: Maximum number of iterations to perform. Default 40.
:param tolerance: Stop iteration if (A - I*eigenvalue) @ v < tolerance.
Default 1e-13.
:param solver: Solver function of the form x = solver(A, b).
By default, use scipy.sparse.spsolve for sparse matrices and
scipy.sparse.bicgstab for general LinearOperator instances.
:return: (eigenvalue, eigenvector)
"""
try:
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_test = operator - sparse.eye(operator.shape[0])
shift = lambda eigval: eigval * sparse.eye(operator.shape[0])
if solver is None:
solver = spalg.spsolve
except TypeError:
shift = lambda eigval: spalg.LinearOperator(shape=operator.shape,
dtype=operator.dtype,
matvec=lambda v: eigval * v)
if solver is None:
solver = lambda A, b: spalg.bicgstab(A, b)[0]
v = guess_vector
v /= norm(v)
for _ in range(iterations):
eigval = v.conj() @ (operator @ v)
if norm(operator @ v - eigval * v) < tolerance:
break
shifted_operator = operator - shift(eigval)
v = solver(shifted_operator, v)
v /= norm(v)
return eigval, v
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def signed_eigensolve(operator: sparse.spmatrix or spalg.LinearOperator,
how_many: int,
negative: bool = False,
) -> Tuple[numpy.ndarray, numpy.ndarray]:
"""
Find the largest-magnitude positive-only (or negative-only) eigenvalues and
eigenvectors of the provided matrix.
:param operator: Matrix to analyze.
:param how_many: How many eigenvalues to find.
:param negative: Whether to find negative-only eigenvalues.
Default False (positive only).
:return: (sorted list of eigenvalues, 2D ndarray of corresponding eigenvectors)
eigenvectors[:, k] corresponds to the k-th eigenvalue
"""
# Use power iteration to estimate the dominant eigenvector
lm_eigval, _ = power_iteration(operator)
'''
Shift by the absolute value of the largest eigenvalue, then find a few of the
largest-magnitude (shifted) eigenvalues. A positive shift ensures that we find the
largest _positive_ eigenvalues, since any negative eigenvalues will be shifted to the
range 0 >= neg_eigval + abs(lm_eigval) > abs(lm_eigval)
'''
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shift = numpy.abs(lm_eigval)
if negative:
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shift *= -1
# Try to combine, use general LinearOperator if we fail
try:
shifted_operator = operator + shift * sparse.eye(operator.shape[0])
except TypeError:
shifted_operator = operator + spalg.LinearOperator(shape=operator.shape,
matvec=lambda v: shift * v)
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shifted_eigenvalues, eigenvectors = spalg.eigs(shifted_operator, which='LM', k=how_many, ncv=50)
eigenvalues = shifted_eigenvalues - shift
k = eigenvalues.argsort()
return eigenvalues[k], eigenvectors[:, k]