QMR Smoothing for Lanczos-Type Product Methods Based on Three-Term Rrecurrences

For the solution of large, sparse, non-Hermitian linear systems, Lanczos-type product methods that are based on the Lanczos three-term recurrence are derived in a systematic way. These methods either square the Lanczos process or combine it with a local minimization of the residual. For them a quasi-minimal residual (QMR) smoothing is proposed that can also be implemented by short-term recurrences. The practical performance of these methods and the QMR smoothing is demonstrated in a number of numerical examples.