A variable preconditioned GCR() method using the GSOR method for singular and rectangular linear systems

The Generalized Conjugate Residual (GCR) method with a variable preconditioning is an efficient method for solving a large sparse linear system A x = 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 such linear systems that the coefficient matrix has diagonal entries of zero or is not square. Therefore, we propose a type of the generalized SOR (GSOR) method. By numerical experiments on the singular linear systems, we demonstrate that the variable preconditioned GCR using GSOR is effective.