Convergence on Gauss-Seidel iterative methods for linear systems with general H-matrices

Abstract

It is well known that as a famous type of iterative methods in numerical linear algebra, Gauss-Seidel iterative methods are convergent for linear systems with strictly or irreducibly diagonally dominant matrices, invertible Hâmatrices (generalized strictly diagonally dominant matrices) and Hermitian positive definite matrices. But, the same is not necessarily true for linear systems with non-strictly diagonally dominant matrices and general Hâmatrices. This paper firstly proposes some necessary and sufficient conditions for convergence on Gauss-Seidel iterative methods to establish several new theoretical results on linear systems with nonstrictly diagonally dominant matrices and general Hâmatrices. Then, the convergence results on preconditioned Gauss-Seidel (PGS) iterative methods for general Hâmatrices are presented. Finally, some numerical examples are given to demonstrate the results obtained in this paper.