A Parallel Version of the Unsymmetric Lanczos Algorithm and its Application to QMR

A new version of the unsymmetric Lanczos algorithm without look-ahead is described combining elements of numerical stability and parallel algorithm design. Firstly, stability is obtained by a coupled two-term procedure that generates Lanczos vectors scaled to unit length. Secondly, the algorithm is derived by making all inner products of a single iteration step independent such that global synchronization on parallel distributed memory computers is reduced. Among the algorithms using the Lanczos process as a major component, the quasi-minimal residual (QMR) method for the solution of systems of linear equations is illustrated by an elegant derivation. The resulting QMR algorithm maintains the favorable properties of the Lanczos algorithm while not increasing computational costs as compared with its corresponding original version