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