A note on the Sassenfeld criterion and its relation to H-matrices

TL;DR

This paper revisits Sassenfeld's 1951 convergence criterion for Gauss-Seidel, revealing its direct connection to H-matrices and providing a new characterization of H-matrices, with implications for iterative solver convergence.

Contribution

It establishes a novel link between Sassenfeld's criterion and H-matrices, offering a new characterization and insights into solver convergence with H-matrix preconditioners.

Findings

Sassenfeld's criterion is directly related to H-matrices.

A new characterization of H-matrices is provided.

Implications for convergence of iterative solvers with H-matrix preconditioners.

Abstract

The starting point of this note is a decades-old yet little-noticed sufficient condition, presented by Sassenfeld in 1951, for the convergence of the classical Gauss-Seidel method. The purpose of the present paper is to shed new light on Sassenfeld's criterion and to demonstrate that it is directly related to H-matrices. In particular, our results yield a new characterization of H-matrices. In addition, the convergence of iterative linear solvers that involve H-matrix preconditioners is briefly discussed.