Communications of the
Korean Mathematical Society CKMS

In 2009, Zheng and Miao [B. Zheng and S.-X. Miao, {\it Two new modified Gauss-Seidel methods for linear system with $M$-matrices}, J. Comput. Appl. Math. {\bf 233} (2009), 922--930] considered the modified Gauss-Seidel method for solving $M$-matrix linear system with the preconditioner $P_{\max}$. In this paper, we consider the modified Gauss-Seidel method for solving the linear system with the generalized preconditioner $P_{\max}(\alpha)$, and study its convergent properties when the coefficient matrix is an $H$-matrix. Numerical experiments are performed with different examples, and the numerical results verify our theoretical analysis.

Keywords: $H$-matrix, preconditioner, modified Gauss-Seidel method, convergence