Convergence of Stationary Iterative Methods for Hermitian Semidefinite Linear Systems and Applications to Schwarz Methods

Abstract

A simple proof is presented of a quite general theorem on the convergence of stationary iterations for solving singular linear systems whose coefficient matrix is Hermitian and positive semidefinite. In this manner, elegant proofs are obtained of some known convergence results, including the necessity of the P-regular splitting result due to Keller, as well as recent results involving generalized inverses. Other generalizations are also presented. These results are then used to analyze the convergence of several versions of algebraic additive and multiplicative Schwarz methods for Hermitian positive semidefinite systems.