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Pure SciPy implementation of Locally Optimal Block Preconditioned Conjugate
Gradient Method (LOBPCG), see
License: BSD
Authors: Robert Cimrman, Andrew Knyazev
Examples in tests directory contributed by Nils Wagner.
from __future__ import division, print_function, absolute_import
import sys
import numpy as np
from numpy.testing import assert_allclose
from scipy._lib.six import xrange
from scipy.linalg import inv, eigh, cho_factor, cho_solve, cholesky
from scipy.sparse.linalg import aslinearoperator, LinearOperator
__all__ = ['lobpcg']
def pause():
# Used only when verbosity level > 10.
def save(ar, fileName):
# Used only when verbosity level > 10.
from numpy import savetxt
savetxt(fileName, ar, precision=8)
def _assert_symmetric(M, rtol=1e-5, atol=1e-8):
assert_allclose(M.T, M, rtol=rtol, atol=atol)
# 21.05.2007, c
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
>>> A = _makeOperator( arrayA, (n, n) )
>>> vectorB = A( vectorX )
if operatorInput is None:
def ident(x):
return x
operator = LinearOperator(expectedShape, ident, matmat=ident)
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."""
gramYBV =, blockVectorV)
tmp = cho_solve(factYBY, gramYBV)
blockVectorV -=, tmp)
def _b_orthonormalize(B, blockVectorV, blockVectorBV=None, retInvR=False):
if blockVectorBV is None:
if B is not None:
blockVectorBV = B(blockVectorV)
blockVectorBV = blockVectorV # Shared data!!!
gramVBV =, blockVectorBV)
gramVBV = cholesky(gramVBV)
gramVBV = inv(gramVBV, overwrite_a=True)
# gramVBV is now R^{-1}.
blockVectorV =, gramVBV)
if B is not None:
blockVectorBV =, gramVBV)
if retInvR:
return blockVectorV, blockVectorBV, gramVBV
return blockVectorV, blockVectorBV
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.
A : {sparse matrix, dense matrix, LinearOperator}
The symmetric linear operator of the problem, usually a
sparse matrix. Often called the "stiffness matrix".
X : array_like
Initial approximation to the k eigenvectors. 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 : array_like, optional
n-by-sizeY matrix of constraints, sizeY < n
The iterations will be performed in the B-orthogonal complement
of the column-space of Y. Y must be full rank.
w : array
Array of k eigenvalues
v : array
An array of k eigenvectors. V has the same shape as X.
Other Parameters
tol : scalar, optional
Solver tolerance (stopping criterion)
by default: tol=n*sqrt(eps)
maxiter : integer, optional
maximum number of iterations
by default: maxiter=min(n,20)
largest : bool, optional
when True, solve for the largest eigenvalues, otherwise the smallest
verbosityLevel : integer, optional
controls solver output. default: verbosityLevel = 0.
retLambdaHistory : boolean, optional
whether to return eigenvalue history
retResidualNormsHistory : boolean, optional
whether to return history of residual norms
Solve A x = lambda B x with constraints and preconditioning.
>>> from scipy.sparse import spdiags, issparse
>>> from scipy.sparse.linalg import lobpcg, LinearOperator
>>> n = 100
>>> vals = [np.arange(n, dtype=np.float64) + 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.]])
>>> 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 -- inverse of A (as an abstract linear operator).
>>> invA = spdiags([1./vals[0]], 0, n, n)
>>> def precond( x ):
... return invA * x
>>> M = LinearOperator(matvec=precond, shape=(n, n), dtype=float)
Here, ``invA`` could of course have been used directly as a preconditioner.
Let us then solve the problem:
>>> eigs, vecs = lobpcg(A, X, Y=Y, M=M, tol=1e-4, maxiter=40, largest=False)
>>> eigs
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.
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 3``m`` 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 5``m``>``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 the LOBPCG code with ``m``=1
and ``n``=10, it should work, though ``n`` is small. The method is intended
for extremely large ``n``/``m``, see e.g., reference [28] in
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.
*Acknowledgements* code was written by Robert Cimrman.
Many thanks belong to Andrew Knyazev, the author of the algorithm,
for lots of advice and support.
.. [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.
.. [2] A. V. Knyazev, I. Lashuk, M. E. Argentati, and E. Ovchinnikov (2007),
Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX)
in hypre and PETSc.
.. [3] A. V. Knyazev's C and MATLAB implementations:
blockVectorX = X
blockVectorY = Y
residualTolerance = tol
maxIterations = maxiter
if blockVectorY is not None:
sizeY = blockVectorY.shape[1]
sizeY = 0
# Block size.
if len(blockVectorX.shape) != 2:
raise ValueError('expected rank-2 array for argument X')
n, sizeX = blockVectorX.shape
if sizeX > n:
raise ValueError('X column dimension exceeds the row dimension')
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.')
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)
eigvals = (0, sizeX-1)
A_dense = A(np.eye(n))
B_dense = None if B is None else B(np.eye(n))
return eigh(A_dense, B_dense, eigvals=eigvals, check_finite=False)
if residualTolerance is None:
residualTolerance = np.sqrt(1e-15) * n
maxIterations = min(n, maxIterations)
if verbosityLevel:
aux = "Solving "
if B is None:
aux += "standard"
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"
if sizeY > 1:
aux += "%d constraints\n\n" % sizeY
aux += "%d constraint\n\n" % sizeY
# Apply constraints to X.
if blockVectorY is not None:
if B is not None:
blockVectorBY = B(blockVectorY)
blockVectorBY = blockVectorY
# gramYBY is a dense array.
gramYBY =, blockVectorBY)
# gramYBY is a Cholesky factor from now on...
gramYBY = cho_factor(gramYBY)
raise ValueError('cannot handle linearly dependent constraints')
_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 =, blockVectorAX)
_lambda, eigBlockVector = eigh(gramXAX, check_finite=False)
ii = np.argsort(_lambda)[:sizeX]
if largest:
ii = ii[::-1]
_lambda = _lambda[ii]
eigBlockVector = np.asarray(eigBlockVector[:,ii])
blockVectorX =, eigBlockVector)
blockVectorAX =, eigBlockVector)
if B is not None:
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
for iterationNumber in xrange(maxIterations):
if verbosityLevel > 0:
print('iteration %d' % iterationNumber)
aux = blockVectorBX * _lambda[np.newaxis,:]
blockVectorR = blockVectorAX - aux
aux = np.sum(blockVectorR.conjugate() * blockVectorR, 0)
residualNorms = np.sqrt(aux)
ii = np.where(residualNorms > residualTolerance, True, False)
activeMask = activeMask & ii
if verbosityLevel > 2:
currentBlockSize = activeMask.sum()
if currentBlockSize != previousBlockSize:
previousBlockSize = currentBlockSize
ident = np.eye(currentBlockSize, dtype=A.dtype)
if currentBlockSize == 0:
if verbosityLevel > 0:
print('current block size:', currentBlockSize)
print('eigenvalue:', _lambda)
print('residual norms:', residualNorms)
if verbosityLevel > 10:
activeBlockVectorR = as2d(blockVectorR[:,activeMask])
if iterationNumber > 0:
activeBlockVectorP = as2d(blockVectorP[:,activeMask])
activeBlockVectorAP = as2d(blockVectorAP[:,activeMask])
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:
gramYBY, blockVectorBY, blockVectorY)
# B-orthonormalize the preconditioned residuals.
aux = _b_orthonormalize(B, activeBlockVectorR)
activeBlockVectorR, activeBlockVectorBR = aux
activeBlockVectorAR = A(activeBlockVectorR)
if iterationNumber > 0:
aux = _b_orthonormalize(B, activeBlockVectorP,
activeBlockVectorBP, retInvR=True)
activeBlockVectorP, activeBlockVectorBP, invR = aux
activeBlockVectorAP =, invR)
# Perform the Rayleigh Ritz Procedure:
# Compute symmetric Gram matrices:
xaw =, activeBlockVectorAR)
waw =, activeBlockVectorAR)
xbw =, activeBlockVectorBR)
if iterationNumber > 0:
xap =, activeBlockVectorAP)
wap =, activeBlockVectorAP)
pap =, activeBlockVectorAP)
xbp =, activeBlockVectorBP)
wbp =, activeBlockVectorBP)
gramA = np.bmat([[np.diag(_lambda), xaw, xap],
[xaw.T, waw, wap],
[xap.T, wap.T, pap]])
gramB = np.bmat([[ident0, xbw, xbp],
[xbw.T, ident, wbp],
[xbp.T, wbp.T, ident]])
gramA = np.bmat([[np.diag(_lambda), xaw],
[xaw.T, waw]])
gramB = np.bmat([[ident0, xbw],
[xbw.T, ident]])
if verbosityLevel > 10:
save(gramA, 'gramA')
save(gramB, 'gramB')
# Solve the generalized eigenvalue problem.
_lambda, eigBlockVector = eigh(gramA, gramB, check_finite=False)
ii = np.argsort(_lambda)[:sizeX]
if largest:
ii = ii[::-1]
if verbosityLevel > 10:
_lambda = _lambda[ii].astype(np.float64)
eigBlockVector = np.asarray(eigBlockVector[:,ii].astype(np.float64))
if verbosityLevel > 10:
print('lambda:', _lambda)
## # Normalize eigenvectors!
## aux = np.sum( eigBlockVector.conjugate() * eigBlockVector, 0 )
## eigVecNorms = np.sqrt( aux )
## eigBlockVector = eigBlockVector / eigVecNorms[np.newaxis,:]
# eigBlockVector, aux = _b_orthonormalize( B, eigBlockVector )
if verbosityLevel > 10:
# Compute Ritz vectors.
if iterationNumber > 0:
eigBlockVectorX = eigBlockVector[:sizeX]
eigBlockVectorR = eigBlockVector[sizeX:sizeX+currentBlockSize]
eigBlockVectorP = eigBlockVector[sizeX+currentBlockSize:]
pp =, eigBlockVectorR)
pp +=, eigBlockVectorP)
app =, eigBlockVectorR)
app +=, eigBlockVectorP)
bpp =, eigBlockVectorR)
bpp +=, eigBlockVectorP)
eigBlockVectorX = eigBlockVector[:sizeX]
eigBlockVectorR = eigBlockVector[sizeX:]
pp =, eigBlockVectorR)
app =, eigBlockVectorR)
bpp =, eigBlockVectorR)
if verbosityLevel > 10:
blockVectorX =, eigBlockVectorX) + pp
blockVectorAX =, eigBlockVectorX) + app
blockVectorBX =, eigBlockVectorX) + bpp
blockVectorP, blockVectorAP, blockVectorBP = pp, app, bpp
aux = blockVectorBX * _lambda[np.newaxis,:]
blockVectorR = blockVectorAX - aux
aux = np.sum(blockVectorR.conjugate() * blockVectorR, 0)
residualNorms = np.sqrt(aux)
if verbosityLevel > 0:
print('final eigenvalue:', _lambda)
print('final residual norms:', residualNorms)
if retLambdaHistory:
if retResidualNormsHistory:
return _lambda, blockVectorX, lambdaHistory, residualNormsHistory
return _lambda, blockVectorX, lambdaHistory
if retResidualNormsHistory:
return _lambda, blockVectorX, residualNormsHistory
return _lambda, blockVectorX