Convergence analysis of the preconditioned Gauss–Seidel method for H-matrices

Abstract

In 1997, Kohno et al. [Toshiyuki Kohno, Hisashi Kotakemori, Hiroshi Niki, Improving the modified Gauss–Seidel method for Z -matrices, Linear Algebra Appl. 267 (1997) 113–123] proved that the convergence rate of the preconditioned Gauss–Seidel method for irreducibly diagonally dominant Z -matrices with a preconditioner I + S α is superior to that of the basic iterative method. In this paper, we present a new preconditioner I + K β which is different from the preconditioner given by Kohno et al. [Toshiyuki Kohno, Hisashi Kotakemori, Hiroshi Niki, Improving the modified Gauss–Seidel method for Z -matrices, Linear Algebra Appl. 267 (1997) 113–123] and prove the convergence theory about two preconditioned iterative methods when the coefficient matrix is an H -matrix. Meanwhile, two novel sufficient conditions for guaranteeing the convergence of the preconditioned iterative methods are given.