A Family of Quasi-Minimal Residual Methods for Nonsymmetric Linear Systems

Abstract

The quasi-minimal residual (QMR) algorithm by Freund, Gutknecht, and Nachtigal [Research Institute forAdvanced Computer Science Tech. Report 91.09, NASA Ames Research Center, Moffett Field, CA; Numer. Math., 60 (1991), pp. 315–339], in addition to its ability to avoid breakdowns, also improves the irregular convergence behavior encountered by the biconjugate gradient (BiCG) method through the incorporation of quasi minimization. This paper studies this quasi-minimal residual approach applied, for simplicity, to the nonlook-ahead version of the QMR method. In particular, a family of quasi-minimal residual methods is defined where on both ends lie the original QMR method and a minimal residual method while lying in between are the variants derived here. These variants are shown to consume a little more computational work per iterations than the original QMR method, but generally give an even better convergence behavior as well as lower iteration counts. The same idea was also applied to a recently proposed transpose-free quasi-minimal residual method. In particular, a few quasi-minimal residual variants of the Van der Vorst BiCGSTAB method are derived. Numerical results are presented comparing the quasi-minimal residual and transpose-free quasi-minimal residual variants derived in this paper.