On the QMR approach for iterative methods including coupled three-term recurrences for solving nonsymmetric linear systems

Abstract

Recently many simple Quasi-Minimal Residual (QMR) approaches have been proposed to improve the convergence behavior of the BI-CG algorithm and its variants (see Chan et al., 1994; Freund, 1993; Freund and Nachtigal, 1991; Freund and Szeto, 1991; Tong, 1994). For using them to obtain improved approximate solutions one needs only to change a few lines in the original algorithms. In most of these approaches the underlying iterative methods to be improved include only two-term recurrences. An exception is the approach of Freund and Nachtigal (1991). In this paper we present a simple but universal QMR approach for constructing QMR variants of any iterative method which includes three-term recurrences. Unified formulas for obtaining improved approximate solutions are derived. The resulting QMR variants can be implemented in a unified manner by adding only a few lines to the original algorithms. Applications of this QMR method to the BICGSTAB2 algorithm of Gutknecht (1993) and the BICGSTAB3 algorithm, which is presented in this paper and is an algorithm with full three-term recurrences, are described. In addition, our QMR approach can also be applied easily to lookhead (i.e., breakdown avoiding) algorithms. Finally, numerical experiments are reported.