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

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. 1. Introduction. In 1950, Lanczos 20] proposed a method for successive reduction of a given, in general non-Hermitian, N N matrix A to tridiagonal form. More precisely, the Lanczos procedure generates a sequence H (n) , n = 1; 2; : : :, of n n tridiagonal matrices which, in a certain sense, approximate A. Furthermore, in exact arithmetic and if no breakdown occurs, the Lanczos method terminates after at most L (N) steps with H (L) a tridiagonal matrix which represents the restriction of A or A