A NEW ADAPTIVE GMRES ALGORITHM FOR ACHIEVING HIGH-ACCURACY

GMRES(k) is widely used for solving nonsymmetric linear systems. Howev er, it is inadequate either when it converges only for k close to the problem size or when numerical error in the modified Gram-Schmidt proc ess used in the GMRES orthogonalization phase dramatically affects the algorithm performance. An adaptive version of GMRES(k) which tunes th e restart value k based on criteria estimating the GMRES convergence r ate for the given problem is proposed here. This adaptive GMRES(k) pro cedure outperforms standard GMRES(k), several other GMRES-like methods , and QMR on actual large scale sparse structural mechanics postbuckli ng and analog circuit simulation problems. There are some applications , such as homotopy methods for high Reynolds number viscous flows, sol id mechanics postbuckling analysis, and analog circuit simulation, whe re very high accuracy in the linear system solutions is essential. In this context, the modified Gram-Schmidt process in GMRES can fail caus ing the entire GMRES iteration to fail. It is shown that the adaptive GMRES(k) with the orthogonalization performed by Householder transform ations succeeds whenever GMRES(k) with the orthogonalization performed by the modified Gram-Schmidt process fails, and the extra cost of com puting Householder transformations is justified for these applications . (C) 1998 John Wiley & Sons, Ltd.