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""" 

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, 

LinAlgError) 

from scipy.sparse.linalg import aslinearoperator 

from scipy.sparse.sputils import bmat 

 

__all__ = ['lobpcg'] 

 

 

def _save(ar, fileName): 

# Used only when verbosity level > 10. 

np.savetxt(fileName, ar) 

 

 

def _report_nonhermitian(M, a, b, name): 

""" 

Report if `M` is not a hermitian matrix given the tolerances `a`, `b`. 

""" 

from scipy.linalg import norm 

 

md = M - M.T.conj() 

 

nmd = norm(md, 1) 

tol = np.spacing(max(10**a, (10**b)*norm(M, 1))) 

if nmd > tol: 

print('matrix %s is not sufficiently Hermitian for a=%d, b=%d:' 

% (name, a, b)) 

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.""" 

gramYBV = np.dot(blockVectorBY.T.conj(), blockVectorV) 

tmp = cho_solve(factYBY, gramYBV) 

blockVectorV -= np.dot(blockVectorY, tmp) 

 

 

def _b_orthonormalize(B, blockVectorV, blockVectorBV=None, retInvR=False): 

if blockVectorBV is None: 

if B is not None: 

blockVectorBV = B(blockVectorV) 

else: 

blockVectorBV = blockVectorV # Shared data!!! 

gramVBV = np.dot(blockVectorV.T.conj(), blockVectorBV) 

gramVBV = cholesky(gramVBV) 

gramVBV = inv(gramVBV, overwrite_a=True) 

# gramVBV is now R^{-1}. 

blockVectorV = np.dot(blockVectorV, gramVBV) 

if B is not None: 

blockVectorBV = np.dot(blockVectorBV, gramVBV) 

else: 

blockVectorBV = None 

 

if retInvR: 

return blockVectorV, blockVectorBV, gramVBV 

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 : 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. 

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 

 

Returns 

------- 

w : array 

Array of k eigenvalues 

v : array 

An array of k eigenvectors. V has the same shape as X. 

lambdas : list of arrays, optional 

The eigenvalue history, if `retLambdaHistory` is True. 

rnorms : list of arrays, optional 

The history of residual norms, if `retResidualNormsHistory` is True. 

 

Examples 

-------- 

 

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.]]) 

 

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 -- 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, 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. 

 

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 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 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. 

 

*Acknowledgements* 

 

lobpcg.py code was written by Robert Cimrman. 

Many thanks belong to Andrew Knyazev, the author of the algorithm, 

for lots of advice and support. 

 

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 

""" 

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.') 

 

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: 

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 = 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 

while iterationNumber < maxIterations: 

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.conjugate() * 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-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 = aux 

activeBlockVectorAP = np.dot(activeBlockVectorAP, invR) 

 

else: 

aux = _b_orthonormalize(B, activeBlockVectorP, retInvR=True) 

activeBlockVectorP, _, invR = aux 

activeBlockVectorAP = np.dot(activeBlockVectorAP, invR) 

 

## 

# Perform the Rayleigh Ritz Procedure: 

# Compute symmetric Gram matrices: 

 

if B is not None: 

xaw = np.dot(blockVectorX.T.conj(), activeBlockVectorAR) 

waw = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAR) 

xbw = np.dot(blockVectorX.T.conj(), activeBlockVectorBR) 

 

if iterationNumber > 0: 

xap = np.dot(blockVectorX.T.conj(), activeBlockVectorAP) 

wap = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAP) 

pap = np.dot(activeBlockVectorP.T.conj(), activeBlockVectorAP) 

xbp = np.dot(blockVectorX.T.conj(), activeBlockVectorBP) 

wbp = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorBP) 

 

gramA = bmat([[np.diag(_lambda), xaw, xap], 

[xaw.T.conj(), waw, wap], 

[xap.T.conj(), wap.T.conj(), pap]]) 

 

gramB = bmat([[ident0, xbw, xbp], 

[xbw.T.conj(), ident, wbp], 

[xbp.T.conj(), wbp.T.conj(), ident]]) 

else: 

gramA = bmat([[np.diag(_lambda), xaw], 

[xaw.T.conj(), waw]]) 

gramB = bmat([[ident0, xbw], 

[xbw.T.conj(), ident]]) 

 

else: 

xaw = np.dot(blockVectorX.T.conj(), activeBlockVectorAR) 

waw = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAR) 

xbw = np.dot(blockVectorX.T.conj(), activeBlockVectorR) 

 

if iterationNumber > 0: 

xap = np.dot(blockVectorX.T.conj(), activeBlockVectorAP) 

wap = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAP) 

pap = np.dot(activeBlockVectorP.T.conj(), activeBlockVectorAP) 

xbp = np.dot(blockVectorX.T.conj(), activeBlockVectorP) 

wbp = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorP) 

 

gramA = bmat([[np.diag(_lambda), xaw, xap], 

[xaw.T.conj(), waw, wap], 

[xap.T.conj(), wap.T.conj(), pap]]) 

 

gramB = bmat([[ident0, xbw, xbp], 

[xbw.T.conj(), ident, wbp], 

[xbp.T.conj(), wbp.T.conj(), ident]]) 

else: 

gramA = bmat([[np.diag(_lambda), xaw], 

[xaw.T.conj(), waw]]) 

gramB = bmat([[ident0, xbw], 

[xbw.T.conj(), ident]]) 

 

if verbosityLevel > 0: 

_report_nonhermitian(gramA, 3, -1, 'gramA') 

_report_nonhermitian(gramB, 3, -1, 'gramB') 

 

if verbosityLevel > 10: 

_save(gramA, 'gramA') 

_save(gramB, 'gramB') 

 

# Solve the generalized eigenvalue problem. 

_lambda, eigBlockVector = eigh(gramA, gramB, check_finite=False) 

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.conjugate() * 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 iterationNumber > 0: 

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 iterationNumber > 0: 

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.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 

else: 

return _lambda, blockVectorX, lambdaHistory 

else: 

if retResidualNormsHistory: 

return _lambda, blockVectorX, residualNormsHistory 

else: 

return _lambda, blockVectorX