The Rate of Convergence of the SOR Method in the Positive Semidefinite Case

Abstract

In this paper, we derive upper bounds that characterize the rate of convergence of the SOR method for solving a linear system of the form $Gx=b$ , where $G$ is a real symmetric positive semidefinite $n\times n$ matrix. The bounds are given in terms of the condition number of $G$ , which is the ratio $\kappa =\alpha /\beta$ , where $\alpha$ is the largest eigenvalue of $G$ and $\beta$ is the smallest nonzero eigenvalue of $G$ . Let $H$ denote the related iteration matrix. Then, since $G$ has a zero eigenvalue, the spectral radius of $H$ equals 1, and the rate of convergence is determined by the size of $\eta$ , the largest eigenvalue of $H$ whose modulus differs from 1. The bound has the form ${\left|\eta \right|}^2\le 1-1/\left(\kappa c\right)$ , where $c=2+{\log }_{2}n.$ The main consequence from this bound is that small condition number forces fast convergence while large condition number allows slow convergence.