A Variant of SOR Method for Singular Linear Systems and its Application to a Variable Preconditioned GCR Method

The Generalized Conjugate Residual (GCR) method with a variable preconditioning is an efficient method for solving a large sparse linear system Ax = b. It has been clarified by some numerical experiments that the Successive Over Relaxation (SOR) method is more effective than Krylov subspace methods such as GCR and ILU(0) preconditioned GCR for performing the variable preconditioning. However, SOR cannot be applied for performing the variable preconditioning, when solving the singular linear system, where the coefficient matrix has diagonal elements of 0 or is rectangular matrix. Therefore, we reconsider the splitting of A, and propose a variant of SOR. By numerical experiments on the singular linear system, we show the efficiency of the variable preconditioned GCR method using the variant of SOR.