scipy.sparse.linalg.lobpcg

scipy.sparse.linalg.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.

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 : 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

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 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 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. 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. 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 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, 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.