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

TL;DR

This paper establishes new necessary and sufficient conditions for the convergence of Gauss-Seidel iterative methods applied to linear systems with general H-matrices, extending known results to broader matrix classes.

Contribution

It introduces novel theoretical conditions for convergence of Gauss-Seidel methods on nonstrictly diagonally dominant and general H-matrices, including preconditioned variants.

Findings

New convergence conditions for nonstrictly diagonally dominant matrices

Convergence results for preconditioned Gauss-Seidel methods on H-matrices

Numerical examples validating theoretical results

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 $H-$matrices (generalized strictly diagonally dominant matrices) and Hermitian positive definite matrices. But, the same is not necessarily true for linear systems with nonstrictly diagonally dominant matrices and general $H-$matrices. 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 $H-$matrices. Then, the convergence results on preconditioned Gauss-Seidel (PGS) iterative methods for general $H-$matrices are presented. Finally, some numerical examples are given to demonstrate the results obtained in this paper.