scipy.sparse.linalg.lobpcg(A, X, B=None, M=None, Y=None, tol=None, maxiter=20, largest=True, verbosityLevel=0, retLambdaHistory=False, retResidualNormsHistory=False)[source]

Solve symmetric partial eigenproblems with optional preconditioning

This function implements the Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG).

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.

Returns :

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 : boolean, 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


If both retLambdaHistory and retResidualNormsHistory are True, the return tuple has the following format (lambda, V, lambda history, residual norms history)