A variable preconditioning using the SOR method for GCR-like methods
Kuniyoshi Abe 1, Shao-Liang Zhang 21 Faculty of Economics and Information, Gifu Shotoku University, Gifu, 500-8288 Japan
2 Graduate School of Engineering, University of Tokyo, Tokyo, 113-8656 Japan
Received by the editors October 15, 2003 and, in revised form, March 22, 2004
We propose a variant of variable preconditioning for Generalized Conjugate Residual (CCR)-like methods. The preconditioning is carried out by roughly solving Az = nu by an iterative method to a certain degree of accuracy instead of computing Kz = nu in a conventional preconditioned algorithm. In our proposal, the number of iterations required for computing Az = nu is changed at each iteration by establishing a stopping criterion. This enables the use of a stationary iterative method when applying different preconditioners. The proposed procedure is incorporated into GCR, and the mathematical convergence is proved. In numerical experiments, we employ the Successive Over-Relaxation (SOR) method for computing Az = nu, and we demonstrate that GCR with the variable preconditioning using SOR is faster and more robust than CCR with an incomplete LU preconditioning, and the FGMRES and GMRESR methods with the variable preconditioning using the Generalized Minimal Residual (GMRES) method. Moreover, we confirm that different preconditioners are applied at each iteration.
AMS subject classifications: 65F10, 65F50, 65N22
Key words: linear systems; generalized conjugate residual method; generalized minimal residual method; variable preconditioning; inner-loop and outerloop
Email: firstname.lastname@example.org (K. Abe), email@example.com (S.-L. Zhang)