Convergence Estimation of Restarted GCR(m) Method Using IDR(s)-SOR Preconditioning

摘要

We consider to solve a linear system of equations Ax = b by Nested Iterative Method. Nested Iterative Method consists of outer and inner iterative methods. Outer iterative method has a role of whole convergence. On the other hand, inner iterative method has a role of solving preconditioned equation of M~(-1)v = z. Namely, we solve actually linear systems of equations as Az = v because of M ≈ A. Recently, the authors proposed IDR(s)-based stationary methods. Then they verified that IDR(s)-based stationary methods are significantly more effective than the conventional iterative methods. In this paper, we propose restarted GCR(m) method with IDR(s)-SOR preconditioning as inner iterative method. Moreover, through numerical experiments, we make clear that the proposed methods have excellent convergence rate.