A Variable Preconditioning Using the Sor Method for Gcr-like Methods

We propose a variant of variable preconditioning for Generalized Conjugate Residual (GCR)-like methods. The preconditioning is carried out by roughly solving Az = v by an iterative method to a certain degree of accuracy instead of computing Kz = v in a conventional preconditioned algorithm. In our proposal, the number of iterations required for computing Az = v 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 = v, and we demonstrate that GCR with the variable preconditioning using SOR is faster and more robust than GCR 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.