lobpcg.m

LOBPCG solves Hermitian partial generalized eigenproblems using preconditioning, competes with eigs

Description

This main function LOBPCG is a version of the preconditioned conjugate gradient method (Algorithm 5.1) described in A. V. Knyazev, Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method, SIAM Journal on Scientific Computing 23 (2001), no. 2, pp. 517-541. http://dx.doi.org/10.1137/S1064827500366124

A C-version of this code is a part of the http://code.google.com/p/blopex/
package and is available, e.g., in SLEPc and HYPRE.

Tested in MATLAB 6.5-7.13 and Octave 3.2.3-3.4.2.

• MATLAB release: MATLAB 7.13 (R2011b)

[Andrew Knyazev]

"I encounter some severe differences in memory requirements comparing lobpcg.m with lobpcg from hypre/ij. Is there a way to use hypre with the same requirements as lobpcg.m?"

High memory requirements in hypre/ij in these tests come from hypre BoomerAMG preconditioning, not from the LOBPCG. For my full answer please see my reply at

http://www.mathworks.com/matlabcentral/answers/9534-memory-requirements-of-lobpcg-matlab-and-hypre-implementation-differences

[Elias]

I encounter some severe differences in memory requirements comparing lobpcg.m with lobpcg from hypre/ij. Is there a way to use hypre with the same requirements as lobpcg.m? For full explanation see

http://www.mathworks.com/matlabcentral/answers/9534-memory-requirements-of-lobpcg-matlab-and-hypre-implementation-differences

[Andrew Knyazev]

"Is there any reason why LOBPCG might not work for generalized eigenvalue problems with large sparse, symmetric matrices..."

Very slow convergence is an expected normal behavior of lobpcg for such a problem, without preconditioning. For detailed explanations and possible solutions, see

http://www.mathworks.com/matlabcentral/answers/9063-lobpcg-returning-incorrect-results-for-large-sparse-symmetric-matrices

[Andrew]

Is there any reason why LOBPCG might not work for generalized eigenvalue problems with large sparse, symmetric matrices (size 70 000 x 70 000, with 4.5 million non-zero values)? It has been very efficient for smaller identical problems (reducing the size of these sparse matrices to 10 000 x 10 000), although hasn't worked when I tried it on a problem of that size. In both cases I used a random matrix as an initial guess for the eigenvectors. see

http://www.mathworks.com/matlabcentral/answers/9063-lobpcg-returning-incorrect-results-for-large-sparse-symmetric-matrices

[Andrew Knyazev]

"One question: Is there any way to compute the lowest eigenpairs above a specific value, as in eigs one can choose a SIGMA to find eigenvalues around it?"

In eigs, the SIGMA-option actually solves the so-called "shift-and-invert" problem, see
http://en.wikipedia.org/wiki/Preconditioner#Spectral_transformations . In LOBPCG, this option is not directly supported, but can be implemented by a user, supplying the corresponding functions to LOBPCG.

[Elias]

Nice work, really fast and very efficient. Since eigs crashes my cluster because of memory requirements, this seems to be a much better choice. One question:
Is there any way to compute the lowest eigenpairs above a specific value, as in eigs one can choose a SIGMA to find eigenvalues around it?
Thanks

[Elias]