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

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+Sα is superior to that of the basic iterative method. In this paper, we present a new preconditioner I+Kβ 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.