You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
716 lines
26 KiB
Python
716 lines
26 KiB
Python
"""
|
|
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. :doi:`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. :arxiv:`0705.2626`
|
|
|
|
.. [3] A. V. Knyazev's C and MATLAB implementations:
|
|
https://github.com/lobpcg/blopex
|
|
"""
|
|
|
|
import numpy as np
|
|
from scipy.linalg import (inv, eigh, cho_factor, cho_solve, cholesky,
|
|
LinAlgError)
|
|
from scipy.sparse.linalg import aslinearoperator
|
|
from numpy import block as bmat
|
|
|
|
__all__ = ['lobpcg']
|
|
|
|
|
|
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=None,
|
|
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 = 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`` [4]_.
|
|
|
|
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. :doi:`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. :arxiv:`0705.2626`
|
|
|
|
.. [3] A. V. Knyazev's C and MATLAB implementations:
|
|
https://bitbucket.org/joseroman/blopex
|
|
|
|
.. [4] S. Yamada, T. Imamura, T. Kano, and M. Machida (2006),
|
|
High-performance computing for exact numerical approaches to
|
|
quantum many-body problems on the earth simulator. In Proceedings
|
|
of the 2006 ACM/IEEE Conference on Supercomputing.
|
|
:doi:`10.1145/1188455.1188504`
|
|
|
|
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
|
|
if maxiter is None:
|
|
maxiter = 20
|
|
|
|
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.')
|
|
|
|
sizeX = min(sizeX, n)
|
|
|
|
if blockVectorY is not None:
|
|
raise NotImplementedError('The dense eigensolver '
|
|
'does not support constraints.')
|
|
|
|
# Define the closed range of indices of eigenvalues to return.
|
|
if largest:
|
|
eigvals = (n - sizeX, n-1)
|
|
else:
|
|
eigvals = (0, sizeX-1)
|
|
|
|
A_dense = A(np.eye(n, dtype=A.dtype))
|
|
B_dense = None if B is None else B(np.eye(n, dtype=B.dtype))
|
|
|
|
vals, vecs = eigh(A_dense, B_dense, eigvals=eigvals,
|
|
check_finite=False)
|
|
if largest:
|
|
# Reverse order to be compatible with eigs() in 'LM' mode.
|
|
vals = vals[::-1]
|
|
vecs = vecs[:, ::-1]
|
|
|
|
return vals, vecs
|
|
|
|
if (residualTolerance is None) or (residualTolerance <= 0.0):
|
|
residualTolerance = np.sqrt(1e-15) * n
|
|
|
|
# Apply constraints to X.
|
|
if blockVectorY is not None:
|
|
|
|
if B is not None:
|
|
blockVectorBY = B(blockVectorY)
|
|
else:
|
|
blockVectorBY = blockVectorY
|
|
|
|
# gramYBY is a dense array.
|
|
gramYBY = np.dot(blockVectorY.T.conj(), blockVectorBY)
|
|
try:
|
|
# gramYBY is a Cholesky factor from now on...
|
|
gramYBY = cho_factor(gramYBY)
|
|
except LinAlgError as e:
|
|
raise ValueError('cannot handle linearly dependent constraints') from e
|
|
|
|
_applyConstraints(blockVectorX, gramYBY, blockVectorBY, blockVectorY)
|
|
|
|
##
|
|
# B-orthonormalize X.
|
|
blockVectorX, blockVectorBX = _b_orthonormalize(B, blockVectorX)
|
|
|
|
##
|
|
# Compute the initial Ritz vectors: solve the eigenproblem.
|
|
blockVectorAX = A(blockVectorX)
|
|
gramXAX = np.dot(blockVectorX.T.conj(), blockVectorAX)
|
|
|
|
_lambda, eigBlockVector = eigh(gramXAX, check_finite=False)
|
|
ii = _get_indx(_lambda, sizeX, largest)
|
|
_lambda = _lambda[ii]
|
|
|
|
eigBlockVector = np.asarray(eigBlockVector[:, ii])
|
|
blockVectorX = np.dot(blockVectorX, eigBlockVector)
|
|
blockVectorAX = np.dot(blockVectorAX, eigBlockVector)
|
|
if B is not None:
|
|
blockVectorBX = np.dot(blockVectorBX, eigBlockVector)
|
|
|
|
##
|
|
# Active index set.
|
|
activeMask = np.ones((sizeX,), dtype=bool)
|
|
|
|
lambdaHistory = [_lambda]
|
|
residualNormsHistory = []
|
|
|
|
previousBlockSize = sizeX
|
|
ident = np.eye(sizeX, dtype=A.dtype)
|
|
ident0 = np.eye(sizeX, dtype=A.dtype)
|
|
|
|
##
|
|
# Main iteration loop.
|
|
|
|
blockVectorP = None # set during iteration
|
|
blockVectorAP = None
|
|
blockVectorBP = None
|
|
|
|
iterationNumber = -1
|
|
restart = True
|
|
explicitGramFlag = False
|
|
while iterationNumber < maxiter:
|
|
iterationNumber += 1
|
|
if verbosityLevel > 0:
|
|
print('iteration %d' % iterationNumber)
|
|
|
|
if B is not None:
|
|
aux = blockVectorBX * _lambda[np.newaxis, :]
|
|
else:
|
|
aux = blockVectorX * _lambda[np.newaxis, :]
|
|
|
|
blockVectorR = blockVectorAX - aux
|
|
|
|
aux = np.sum(blockVectorR.conj() * blockVectorR, 0)
|
|
residualNorms = np.sqrt(aux)
|
|
|
|
residualNormsHistory.append(residualNorms)
|
|
|
|
ii = np.where(residualNorms > residualTolerance, True, False)
|
|
activeMask = activeMask & ii
|
|
if verbosityLevel > 2:
|
|
print(activeMask)
|
|
|
|
currentBlockSize = activeMask.sum()
|
|
if currentBlockSize != previousBlockSize:
|
|
previousBlockSize = currentBlockSize
|
|
ident = np.eye(currentBlockSize, dtype=A.dtype)
|
|
|
|
if currentBlockSize == 0:
|
|
break
|
|
|
|
if verbosityLevel > 0:
|
|
print('current block size:', currentBlockSize)
|
|
print('eigenvalue:', _lambda)
|
|
print('residual norms:', residualNorms)
|
|
if verbosityLevel > 10:
|
|
print(eigBlockVector)
|
|
|
|
activeBlockVectorR = _as2d(blockVectorR[:, activeMask])
|
|
|
|
if iterationNumber > 0:
|
|
activeBlockVectorP = _as2d(blockVectorP[:, activeMask])
|
|
activeBlockVectorAP = _as2d(blockVectorAP[:, activeMask])
|
|
if B is not None:
|
|
activeBlockVectorBP = _as2d(blockVectorBP[:, activeMask])
|
|
|
|
if M is not None:
|
|
# Apply preconditioner T to the active residuals.
|
|
activeBlockVectorR = M(activeBlockVectorR)
|
|
|
|
##
|
|
# Apply constraints to the preconditioned residuals.
|
|
if blockVectorY is not None:
|
|
_applyConstraints(activeBlockVectorR,
|
|
gramYBY, blockVectorBY, blockVectorY)
|
|
|
|
##
|
|
# B-orthogonalize the preconditioned residuals to X.
|
|
if B is not None:
|
|
activeBlockVectorR = activeBlockVectorR - np.matmul(blockVectorX,
|
|
np.matmul(blockVectorBX.T.conj(),
|
|
activeBlockVectorR))
|
|
else:
|
|
activeBlockVectorR = activeBlockVectorR - np.matmul(blockVectorX,
|
|
np.matmul(blockVectorX.T.conj(),
|
|
activeBlockVectorR))
|
|
|
|
##
|
|
# B-orthonormalize the preconditioned residuals.
|
|
aux = _b_orthonormalize(B, activeBlockVectorR)
|
|
activeBlockVectorR, activeBlockVectorBR = aux
|
|
|
|
activeBlockVectorAR = A(activeBlockVectorR)
|
|
|
|
if iterationNumber > 0:
|
|
if B is not None:
|
|
aux = _b_orthonormalize(B, activeBlockVectorP,
|
|
activeBlockVectorBP, retInvR=True)
|
|
activeBlockVectorP, activeBlockVectorBP, invR, normal = aux
|
|
else:
|
|
aux = _b_orthonormalize(B, activeBlockVectorP, retInvR=True)
|
|
activeBlockVectorP, _, invR, normal = aux
|
|
# Function _b_orthonormalize returns None if Cholesky fails
|
|
if activeBlockVectorP is not None:
|
|
activeBlockVectorAP = activeBlockVectorAP / normal
|
|
activeBlockVectorAP = np.dot(activeBlockVectorAP, invR)
|
|
restart = False
|
|
else:
|
|
restart = True
|
|
|
|
##
|
|
# Perform the Rayleigh Ritz Procedure:
|
|
# Compute symmetric Gram matrices:
|
|
|
|
if activeBlockVectorAR.dtype == 'float32':
|
|
myeps = 1
|
|
elif activeBlockVectorR.dtype == 'float32':
|
|
myeps = 1e-4
|
|
else:
|
|
myeps = 1e-8
|
|
|
|
if residualNorms.max() > myeps and not explicitGramFlag:
|
|
explicitGramFlag = False
|
|
else:
|
|
# Once explicitGramFlag, forever explicitGramFlag.
|
|
explicitGramFlag = True
|
|
|
|
# Shared memory assingments to simplify the code
|
|
if B is None:
|
|
blockVectorBX = blockVectorX
|
|
activeBlockVectorBR = activeBlockVectorR
|
|
if not restart:
|
|
activeBlockVectorBP = activeBlockVectorP
|
|
|
|
# Common submatrices:
|
|
gramXAR = np.dot(blockVectorX.T.conj(), activeBlockVectorAR)
|
|
gramRAR = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAR)
|
|
|
|
if explicitGramFlag:
|
|
gramRAR = (gramRAR + gramRAR.T.conj())/2
|
|
gramXAX = np.dot(blockVectorX.T.conj(), blockVectorAX)
|
|
gramXAX = (gramXAX + gramXAX.T.conj())/2
|
|
gramXBX = np.dot(blockVectorX.T.conj(), blockVectorBX)
|
|
gramRBR = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorBR)
|
|
gramXBR = np.dot(blockVectorX.T.conj(), activeBlockVectorBR)
|
|
else:
|
|
gramXAX = np.diag(_lambda)
|
|
gramXBX = ident0
|
|
gramRBR = ident
|
|
gramXBR = np.zeros((sizeX, currentBlockSize), dtype=A.dtype)
|
|
|
|
def _handle_gramA_gramB_verbosity(gramA, gramB):
|
|
if verbosityLevel > 0:
|
|
_report_nonhermitian(gramA, 'gramA')
|
|
_report_nonhermitian(gramB, 'gramB')
|
|
if verbosityLevel > 10:
|
|
# Note: not documented, but leave it in here for now
|
|
np.savetxt('gramA.txt', gramA)
|
|
np.savetxt('gramB.txt', gramB)
|
|
|
|
if not restart:
|
|
gramXAP = np.dot(blockVectorX.T.conj(), activeBlockVectorAP)
|
|
gramRAP = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAP)
|
|
gramPAP = np.dot(activeBlockVectorP.T.conj(), activeBlockVectorAP)
|
|
gramXBP = np.dot(blockVectorX.T.conj(), activeBlockVectorBP)
|
|
gramRBP = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorBP)
|
|
if explicitGramFlag:
|
|
gramPAP = (gramPAP + gramPAP.T.conj())/2
|
|
gramPBP = np.dot(activeBlockVectorP.T.conj(),
|
|
activeBlockVectorBP)
|
|
else:
|
|
gramPBP = ident
|
|
|
|
gramA = bmat([[gramXAX, gramXAR, gramXAP],
|
|
[gramXAR.T.conj(), gramRAR, gramRAP],
|
|
[gramXAP.T.conj(), gramRAP.T.conj(), gramPAP]])
|
|
gramB = bmat([[gramXBX, gramXBR, gramXBP],
|
|
[gramXBR.T.conj(), gramRBR, gramRBP],
|
|
[gramXBP.T.conj(), gramRBP.T.conj(), gramPBP]])
|
|
|
|
_handle_gramA_gramB_verbosity(gramA, gramB)
|
|
|
|
try:
|
|
_lambda, eigBlockVector = eigh(gramA, gramB,
|
|
check_finite=False)
|
|
except LinAlgError:
|
|
# try again after dropping the direction vectors P from RR
|
|
restart = True
|
|
|
|
if restart:
|
|
gramA = bmat([[gramXAX, gramXAR],
|
|
[gramXAR.T.conj(), gramRAR]])
|
|
gramB = bmat([[gramXBX, gramXBR],
|
|
[gramXBR.T.conj(), gramRBR]])
|
|
|
|
_handle_gramA_gramB_verbosity(gramA, gramB)
|
|
|
|
try:
|
|
_lambda, eigBlockVector = eigh(gramA, gramB,
|
|
check_finite=False)
|
|
except LinAlgError as e:
|
|
raise ValueError('eigh has failed in lobpcg iterations') from e
|
|
|
|
ii = _get_indx(_lambda, sizeX, largest)
|
|
if verbosityLevel > 10:
|
|
print(ii)
|
|
print(_lambda)
|
|
|
|
_lambda = _lambda[ii]
|
|
eigBlockVector = eigBlockVector[:, ii]
|
|
|
|
lambdaHistory.append(_lambda)
|
|
|
|
if verbosityLevel > 10:
|
|
print('lambda:', _lambda)
|
|
# # Normalize eigenvectors!
|
|
# aux = np.sum( eigBlockVector.conj() * eigBlockVector, 0 )
|
|
# eigVecNorms = np.sqrt( aux )
|
|
# eigBlockVector = eigBlockVector / eigVecNorms[np.newaxis, :]
|
|
# eigBlockVector, aux = _b_orthonormalize( B, eigBlockVector )
|
|
|
|
if verbosityLevel > 10:
|
|
print(eigBlockVector)
|
|
|
|
# Compute Ritz vectors.
|
|
if B is not None:
|
|
if not restart:
|
|
eigBlockVectorX = eigBlockVector[:sizeX]
|
|
eigBlockVectorR = eigBlockVector[sizeX:sizeX+currentBlockSize]
|
|
eigBlockVectorP = eigBlockVector[sizeX+currentBlockSize:]
|
|
|
|
pp = np.dot(activeBlockVectorR, eigBlockVectorR)
|
|
pp += np.dot(activeBlockVectorP, eigBlockVectorP)
|
|
|
|
app = np.dot(activeBlockVectorAR, eigBlockVectorR)
|
|
app += np.dot(activeBlockVectorAP, eigBlockVectorP)
|
|
|
|
bpp = np.dot(activeBlockVectorBR, eigBlockVectorR)
|
|
bpp += np.dot(activeBlockVectorBP, eigBlockVectorP)
|
|
else:
|
|
eigBlockVectorX = eigBlockVector[:sizeX]
|
|
eigBlockVectorR = eigBlockVector[sizeX:]
|
|
|
|
pp = np.dot(activeBlockVectorR, eigBlockVectorR)
|
|
app = np.dot(activeBlockVectorAR, eigBlockVectorR)
|
|
bpp = np.dot(activeBlockVectorBR, eigBlockVectorR)
|
|
|
|
if verbosityLevel > 10:
|
|
print(pp)
|
|
print(app)
|
|
print(bpp)
|
|
|
|
blockVectorX = np.dot(blockVectorX, eigBlockVectorX) + pp
|
|
blockVectorAX = np.dot(blockVectorAX, eigBlockVectorX) + app
|
|
blockVectorBX = np.dot(blockVectorBX, eigBlockVectorX) + bpp
|
|
|
|
blockVectorP, blockVectorAP, blockVectorBP = pp, app, bpp
|
|
|
|
else:
|
|
if not restart:
|
|
eigBlockVectorX = eigBlockVector[:sizeX]
|
|
eigBlockVectorR = eigBlockVector[sizeX:sizeX+currentBlockSize]
|
|
eigBlockVectorP = eigBlockVector[sizeX+currentBlockSize:]
|
|
|
|
pp = np.dot(activeBlockVectorR, eigBlockVectorR)
|
|
pp += np.dot(activeBlockVectorP, eigBlockVectorP)
|
|
|
|
app = np.dot(activeBlockVectorAR, eigBlockVectorR)
|
|
app += np.dot(activeBlockVectorAP, eigBlockVectorP)
|
|
else:
|
|
eigBlockVectorX = eigBlockVector[:sizeX]
|
|
eigBlockVectorR = eigBlockVector[sizeX:]
|
|
|
|
pp = np.dot(activeBlockVectorR, eigBlockVectorR)
|
|
app = np.dot(activeBlockVectorAR, eigBlockVectorR)
|
|
|
|
if verbosityLevel > 10:
|
|
print(pp)
|
|
print(app)
|
|
|
|
blockVectorX = np.dot(blockVectorX, eigBlockVectorX) + pp
|
|
blockVectorAX = np.dot(blockVectorAX, eigBlockVectorX) + app
|
|
|
|
blockVectorP, blockVectorAP = pp, app
|
|
|
|
if B is not None:
|
|
aux = blockVectorBX * _lambda[np.newaxis, :]
|
|
|
|
else:
|
|
aux = blockVectorX * _lambda[np.newaxis, :]
|
|
|
|
blockVectorR = blockVectorAX - aux
|
|
|
|
aux = np.sum(blockVectorR.conj() * blockVectorR, 0)
|
|
residualNorms = np.sqrt(aux)
|
|
|
|
# Future work: Need to add Postprocessing here:
|
|
# Making sure eigenvectors "exactly" satisfy the blockVectorY constrains?
|
|
# Making sure eigenvecotrs are "exactly" othonormalized by final "exact" RR
|
|
# Computing the actual true residuals
|
|
|
|
if verbosityLevel > 0:
|
|
print('final eigenvalue:', _lambda)
|
|
print('final residual norms:', residualNorms)
|
|
|
|
if retLambdaHistory:
|
|
if retResidualNormsHistory:
|
|
return _lambda, blockVectorX, lambdaHistory, residualNormsHistory
|
|
else:
|
|
return _lambda, blockVectorX, lambdaHistory
|
|
else:
|
|
if retResidualNormsHistory:
|
|
return _lambda, blockVectorX, residualNormsHistory
|
|
else:
|
|
return _lambda, blockVectorX
|