Look-Ahead Procedures for Lanczos-Type Product Methods Based on Three-Term Lanczos Recurrences

Abstract

Lanczos-type product methods for the solution of large sparse non-Hermitian linear systems either square the Lanczos process or combine it with a local minimization of the residual. They inherit from the underlying Lanczos process the danger of breakdown. For various Lanczos-type product methods that are based on the Lanczos three-term recurrence, look-ahead versions are presented which avoid such breakdowns or near-breakdowns at the cost of a small computational overhead. Different look-ahead strategies are discussed and their efficiency is demonstrated by several numerical examples.