The simultaneous computation of a few of the algebraically largest and smallest eigenvalues of a large, sparse, symmetric matrix

Abstract

A heuristic argument and supporting numerical results are given to demonstrate that a block Lanczos procedure can be used to compute simultaneously a few of the algebraically largest and smallest eigenvalues and a corresponding eigenspace of a large, sparse, symmetric matrix A. This block procedure can be used, for example, to compute appropriate parameters for iterative schemes used in solving the equation Ax=b. Moreover, if there exists an efficient method for repeatedly solving the equation (A-σI)X=B, this procedure can be used to determine the interior eigenvalues (and corresponding eigenvectors) of A closest to σ. © 1978 BIT Foundations.