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.
"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
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
"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
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
"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.
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