ON THE QMR APPROACH FOR ITERATIVE METHODS INCLUDING COUPLED 3-TERM RECURRENCES FOR SOLVING NONSYMMETRIC LINEAR-SYSTEMS

Recently many simple Quasi-Minimal Residual (QMR) approaches have been proposed to improve the convergence behavior of the BI-CG algorithm a nd its variants (see Chan et al., 1993; Freund, 1993; Freund and Nacht igal, 1991; Freund and Szeto, 1991;Tong, 1993). For using them to obta in improved approximate solutions one needs only to change a few lines in the original algorithms. In most of these approaches the underlyin g iterative methods to be improved include only two-term recurrences. An exception is the approach of Freund and Nachtigal (1991). In this p aper we present a simple but universal QMR approach for constructing Q MR variants of any iterative method which includes three-term recurren ces. Unified formulas for obtaining improved approximate solutions are derived. The resulting QMR variants can be implemented in a unified m anner by adding only a few lines to the original algorithms. Applicati ons of this QMR method to the BICGSTAB2 algorithm of Gutknecht (1993) and the BICGSTAB3 algorithm, which is presented in this paper and is a n algorithm with full three-term recurrences, are described. In additi on, our QMR approach can also be applied easily to lookhead (i.e., bre akdown avoiding) algorithms. Finally, numerical experiments are report ed. (C) 1998 IMACS/Elsevier Science B.V.