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. An implementation of a look-ahead version of the Lanczos algorithm is presented 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.