On quasi-minimal residual approach of iterative algorithms for solving nonsymmetric linear systems

Abstract

Many iterative algorithms, for example the BCG algorithm, the CGS algorithm etc., for solving nonsymmetric linear systems have the erratic convergence behavior. Recently, Freund[6] proposed a BCG-like approach, the Quasi-minimal residual (QMR) method, that remedies this problem for BCG and produces smooth convergence curves.The QMR approach is also applied to CGS and BI-CGSTAB to obtain smoothly convergent variants of these algorithms [7,5,12].In this paper, we propose a simple but universal QMR approach which can be applied with unified manner to any iterative algorithm to construct smoothly convergent variants, provided this algorithm includes two-term recurrence for the approximate solutions. The resulting QMR algorithms can be implemented very easily by changing only a few lines in the original iterative algorithm. We compare the performance of our QMR approach with that of other QMR methods presented in [5],[7] and [12]. Finally, numerical examples are given.