An Implementation of the Look-Ahead Lanczos Algorithm for Non-Hermitian Matrices

Abstract

. The nonsymmetric Lanczos method can be used to compute eigenvalues of large sparse non-Hermitian matrices or to solve large sparse non-Hermitian linear systems. However, the original Lanczos algorithm is susceptible to possible breakdowns and potential instabilities. We present an implementation of a look-ahead version of the Lanczos algorithm that---except for the very special situation of an incurable breakdown--- overcomes these problems by skipping over those steps in which a breakdown or near-breakdown would occur in the standard process. The proposed algorithm can handle look-ahead steps of any length and requires the same number of matrix-vector products and inner products as the standard Lanczos process without look-ahead. Key words. Lanczos method, orthogonal polynomials, look-ahead steps, eigenvalue problems, iterative methods, non-Hermitian matrices, sparse linear systems AMS(MOS) subject classifications. 65F15, 65F10 1. Introduction. In 1950, Lanczos [20] proposed a me..