"""
scikit-learn copy of scipy/sparse/linalg/eigen/lobpcg/lobpcg.py v1.3.0
to be deleted after scipy 1.3.0 becomes a dependency in scikit-lean
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG).
References
----------
.. [1] A. V. Knyazev (2001),
Toward the Optimal Preconditioned Eigensolver: Locally Optimal
Block Preconditioned Conjugate Gradient Method.
SIAM Journal on Scientific Computing 23, no. 2,
pp. 517-541. http://dx.doi.org/10.1137/S1064827500366124
.. [2] A. V. Knyazev, I. Lashuk, M. E. Argentati, and E. Ovchinnikov (2007),
Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX)
in hypre and PETSc. https://arxiv.org/abs/0705.2626
.. [3] A. V. Knyazev's C and MATLAB implementations:
https://bitbucket.org/joseroman/blopex
"""
from __future__ import division, print_function, absolute_import
import numpy as np
from scipy.linalg import (inv, eigh, cho_factor, cho_solve, cholesky, orth,
LinAlgError)
from scipy.sparse.linalg import aslinearoperator
__all__ = ['lobpcg']
def bmat(*args, **kwargs):
import warnings
with warnings.catch_warnings(record=True):
warnings.filterwarnings(
'ignore', '.*the matrix subclass is not the recommended way.*')
return np.bmat(*args, **kwargs)
def _save(ar, fileName):
# Used only when verbosity level > 10.
np.savetxt(fileName, ar)
def _report_nonhermitian(M, name):
"""
Report if `M` is not a hermitian matrix given its type.
"""
from scipy.linalg import norm
md = M - M.T.conj()
nmd = norm(md, 1)
tol = 10 * np.finfo(M.dtype).eps
tol = max(tol, tol * norm(M, 1))
if nmd > tol:
print('matrix %s of the type %s is not sufficiently Hermitian:'
% (name, M.dtype))
print('condition: %.e < %e' % (nmd, tol))
def _as2d(ar):
"""
If the input array is 2D return it, if it is 1D, append a dimension,
making it a column vector.
"""
if ar.ndim == 2:
return ar
else: # Assume 1!
aux = np.array(ar, copy=False)
aux.shape = (ar.shape[0], 1)
return aux
def _makeOperator(operatorInput, expectedShape):
"""Takes a dense numpy array or a sparse matrix or
a function and makes an operator performing matrix * blockvector
products."""
if operatorInput is None:
return None
else:
operator = aslinearoperator(operatorInput)
if operator.shape != expectedShape:
raise ValueError('operator has invalid shape')
return operator
def _applyConstraints(blockVectorV, factYBY, blockVectorBY, blockVectorY):
"""Changes blockVectorV in place."""
YBV = np.dot(blockVectorBY.T.conj(), blockVectorV)
tmp = cho_solve(factYBY, YBV)
blockVectorV -= np.dot(blockVectorY, tmp)
def _b_orthonormalize(B, blockVectorV, blockVectorBV=None, retInvR=False):
"""B-orthonormalize the given block vector using Cholesky."""
normalization = blockVectorV.max(axis=0)+np.finfo(blockVectorV.dtype).eps
blockVectorV = blockVectorV / normalization
if blockVectorBV is None:
if B is not None:
blockVectorBV = B(blockVectorV)
else:
blockVectorBV = blockVectorV # Shared data!!!
else:
blockVectorBV = blockVectorBV / normalization
VBV = np.matmul(blockVectorV.T.conj(), blockVectorBV)
try:
# VBV is a Cholesky factor from now on...
VBV = cholesky(VBV, overwrite_a=True)
VBV = inv(VBV, overwrite_a=True)
blockVectorV = np.matmul(blockVectorV, VBV)
# blockVectorV = (cho_solve((VBV.T, True), blockVectorV.T)).T
if B is not None:
blockVectorBV = np.matmul(blockVectorBV, VBV)
# blockVectorBV = (cho_solve((VBV.T, True), blockVectorBV.T)).T
else:
blockVectorBV = None
except LinAlgError:
# raise ValueError('Cholesky has failed')
blockVectorV = None
blockVectorBV = None
VBV = None
if retInvR:
return blockVectorV, blockVectorBV, VBV, normalization
else:
return blockVectorV, blockVectorBV
def _get_indx(_lambda, num, largest):
"""Get `num` indices into `_lambda` depending on `largest` option."""
ii = np.argsort(_lambda)
if largest:
ii = ii[:-num-1:-1]
else:
ii = ii[:num]
return ii
def lobpcg(A, X,
B=None, M=None, Y=None,
tol=None, maxiter=20,
largest=True, verbosityLevel=0,
retLambdaHistory=False, retResidualNormsHistory=False):
"""Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG)
LOBPCG is a preconditioned eigensolver for large symmetric positive
definite (SPD) generalized eigenproblems.
Parameters
----------
A : {sparse matrix, dense matrix, LinearOperator}
The symmetric linear operator of the problem, usually a
sparse matrix. Often called the "stiffness matrix".
X : ndarray, float32 or float64
Initial approximation to the ``k`` eigenvectors (non-sparse). If `A`
has ``shape=(n,n)`` then `X` should have shape ``shape=(n,k)``.
B : {dense matrix, sparse matrix, LinearOperator}, optional
The right hand side operator in a generalized eigenproblem.
By default, ``B = Identity``. Often called the "mass matrix".
M : {dense matrix, sparse matrix, LinearOperator}, optional
Preconditioner to `A`; by default ``M = Identity``.
`M` should approximate the inverse of `A`.
Y : ndarray, float32 or float64, optional
n-by-sizeY matrix of constraints (non-sparse), sizeY < n
The iterations will be performed in the B-orthogonal complement
of the column-space of Y. Y must be full rank.
tol : scalar, optional
Solver tolerance (stopping criterion).
The default is ``tol=n*sqrt(eps)``.
maxiter : int, optional
Maximum number of iterations. The default is ``maxiter=min(n, 20)``.
largest : bool, optional
When True, solve for the largest eigenvalues, otherwise the smallest.
verbosityLevel : int, optional
Controls solver output. The default is ``verbosityLevel=0``.
retLambdaHistory : bool, optional
Whether to return eigenvalue history. Default is False.
retResidualNormsHistory : bool, optional
Whether to return history of residual norms. Default is False.
Returns
-------
w : ndarray
Array of ``k`` eigenvalues
v : ndarray
An array of ``k`` eigenvectors. `v` has the same shape as `X`.
lambdas : list of ndarray, optional
The eigenvalue history, if `retLambdaHistory` is True.
rnorms : list of ndarray, optional
The history of residual norms, if `retResidualNormsHistory` is True.
Notes
-----
If both ``retLambdaHistory`` and ``retResidualNormsHistory`` are True,
the return tuple has the following format
``(lambda, V, lambda history, residual norms history)``.
In the following ``n`` denotes the matrix size and ``m`` the number
of required eigenvalues (smallest or largest).
The LOBPCG code internally solves eigenproblems of the size ``3m`` on every
iteration by calling the "standard" dense eigensolver, so if ``m`` is not
small enough compared to ``n``, it does not make sense to call the LOBPCG
code, but rather one should use the "standard" eigensolver, e.g. numpy or
scipy function in this case.
If one calls the LOBPCG algorithm for ``5m > n``, it will most likely break
internally, so the code tries to call the standard function instead.
It is not that ``n`` should be large for the LOBPCG to work, but rather the
ratio ``n / m`` should be large. It you call LOBPCG with ``m=1``
and ``n=10``, it works though ``n`` is small. The method is intended
for extremely large ``n / m``, see e.g., reference [28] in
https://arxiv.org/abs/0705.2626
The convergence speed depends basically on two factors:
1. How well relatively separated the seeking eigenvalues are from the rest
of the eigenvalues. One can try to vary ``m`` to make this better.
2. How well conditioned the problem is. This can be changed by using proper
preconditioning. For example, a rod vibration test problem (under tests
directory) is ill-conditioned for large ``n``, so convergence will be
slow, unless efficient preconditioning is used. For this specific
problem, a good simple preconditioner function would be a linear solve
for `A`, which is easy to code since A is tridiagonal.
References
----------
.. [1] A. V. Knyazev (2001),
Toward the Optimal Preconditioned Eigensolver: Locally Optimal
Block Preconditioned Conjugate Gradient Method.
SIAM Journal on Scientific Computing 23, no. 2,
pp. 517-541. http://dx.doi.org/10.1137/S1064827500366124
.. [2] A. V. Knyazev, I. Lashuk, M. E. Argentati, and E. Ovchinnikov
(2007), Block Locally Optimal Preconditioned Eigenvalue Xolvers
(BLOPEX) in hypre and PETSc. https://arxiv.org/abs/0705.2626
.. [3] A. V. Knyazev's C and MATLAB implementations:
https://bitbucket.org/joseroman/blopex
Examples
--------
Solve ``A x = lambda x`` with constraints and preconditioning.
>>> import numpy as np
>>> from scipy.sparse import spdiags, issparse
>>> from scipy.sparse.linalg import lobpcg, LinearOperator
>>> n = 100
>>> vals = np.arange(1, n + 1)
>>> A = spdiags(vals, 0, n, n)
>>> A.toarray()
array([[ 1., 0., 0., ..., 0., 0., 0.],
[ 0., 2., 0., ..., 0., 0., 0.],
[ 0., 0., 3., ..., 0., 0., 0.],
...,
[ 0., 0., 0., ..., 98., 0., 0.],
[ 0., 0., 0., ..., 0., 99., 0.],
[ 0., 0., 0., ..., 0., 0., 100.]])
Constraints:
>>> Y = np.eye(n, 3)
Initial guess for eigenvectors, should have linearly independent
columns. Column dimension = number of requested eigenvalues.
>>> X = np.random.rand(n, 3)
Preconditioner in the inverse of A in this example:
>>> invA = spdiags([1./vals], 0, n, n)
The preconditiner must be defined by a function:
>>> def precond( x ):
... return invA @ x
The argument x of the preconditioner function is a matrix inside `lobpcg`,
thus the use of matrix-matrix product ``@``.
The preconditioner function is passed to lobpcg as a `LinearOperator`:
>>> M = LinearOperator(matvec=precond, matmat=precond,
... shape=(n, n), dtype=float)
Let us now solve the eigenvalue problem for the matrix A:
>>> eigenvalues, _ = lobpcg(A, X, Y=Y, M=M, largest=False)
>>> eigenvalues
array([4., 5., 6.])
Note that the vectors passed in Y are the eigenvectors of the 3 smallest
eigenvalues. The results returned are orthogonal to those.
"""
blockVectorX = X
blockVectorY = Y
residualTolerance = tol
maxIterations = maxiter
if blockVectorY is not None:
sizeY = blockVectorY.shape[1]
else:
sizeY = 0
# Block size.
if len(blockVectorX.shape) != 2:
raise ValueError('expected rank-2 array for argument X')
n, sizeX = blockVectorX.shape
if verbosityLevel:
aux = "Solving "
if B is None:
aux += "standard"
else:
aux += "generalized"
aux += " eigenvalue problem with"
if M is None:
aux += "out"
aux += " preconditioning\n\n"
aux += "matrix size %d\n" % n
aux += "block size %d\n\n" % sizeX
if blockVectorY is None:
aux += "No constraints\n\n"
else:
if sizeY > 1:
aux += "%d constraints\n\n" % sizeY
else:
aux += "%d constraint\n\n" % sizeY
print(aux)
A = _makeOperator(A, (n, n))
B = _makeOperator(B, (n, n))
M = _makeOperator(M, (n, n))
if (n - sizeY) < (5 * sizeX):
# warn('The problem size is small compared to the block size.' \
# ' Using dense eigensolver instead of LOBPCG.')
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