RESTARTING TECHNIQUES FOR THE LANCZOS ALGORITHM AND THEIR IMPLEMENTATION IN PARALLEL COMPUTING ENVIRONMENTS: ARCHITECTURAL INFLUENCES

Abstract

The Lanczos algorithm is one of the principal algorithms for the computation of a few of the extreme eigenvalues and their corresponding eigenvectors of very large, sparse, symmetric matrices and is currently the subject of intense research. In this paper two explicit restart techniques for the algorithm are proposed and their suitability for implementation in two different parallel environments is explored. Both techniques adopt a deflation approach in which exactly one eigenpair at a time is computed. Thus, p eigenpairs are computed using exactly p applications of the chosen Lanczos method. Newly generated Lanczos vectors are orthogona-lized with respect to all previously converged eigenvectors. Each of the techniques has been embedded in a version of the Lanczos algorithm which incorporates full reorthogonalization of the Lanczos vectors as well as a new convergence monitoring routine. Both of the modified Lanczos algorithms have been implemented in a distributed memory SIMD environment and in a shared memory MIMD environment. The execution time efficiency of the two algorithms in each environment for the solution of a variety of matrices selected from the Harwell-Boeing collection of sparse matrices is presented and their performances are compared with those of Sorensen's implicit state-of-the-art routine taken from the ARPACK library. The experiments show that, in an environment in which matrix-vector products may be executed very efficiently, the explicit methods often significantly outperform the implicit method. This result points to the desirability of selecting implementations of algorithms built (as far as possible) from operations which may be efficiently performed in the given parallel environment.