The parallel computation of partial eigensolutions of large matrices on a massively parallel processor

Abstract

The computation of a few eigenvalues and their corresponding eigenvectors of large, usually sparse, real symmetric matrices is often required in the determination of the solution of many of the important problems encountered in scientific and engineering applications. A common characteristic of two distinct classes of iterative methods for the computation of such eigenvalues and eigenvectors is the construction of a sequence of subspaces which contains, in the limit, the desired eigenvectors. In this paper two algorithms, one from each class, for the parallel computation of a few extreme eigenvalues and their associated eigenvectors of large symmetric matrices are discussed. The first algorithm is a simultaneous iteration method in which the subspaces are of a constant dimension; the second is the Lanczos algorithm in which the subspaces increase in dimension. Modified versions of each of the algorithms are proposed and implemented on an MPP Connection Machine CM-200 with 8K processors. A comparative evaluation of the efficiency of the most efficient version of each of the two algorithms for a variety of different types of matrices of maximum order 11,948 is also presented.