A comparison of the Extrapolated Successive Overrelaxation and the Preconditioned Simultaneous Displacement methods for augmented linear systems

TL;DR

This paper compares two preconditioned iterative methods, GMESOR and GMPSD, for solving large sparse augmented linear systems, analyzing their convergence and optimal parameters.

Contribution

It introduces and analyzes the GMESOR and GMPSD methods, providing convergence conditions and a geometric approach for optimal parameter selection.

Findings

Both methods achieve the same convergence rate.

Numerical results confirm theoretical convergence predictions.

Optimal parameters can be determined geometrically.

Abstract

In this paper we study the impact of two types of preconditioning on the numerical solution of large sparse augmented linear systems. The first preconditioning matrix is the lower triangular part whereas the second is the product of the lower triangular part with the upper triangular part of the augmented system's coefficient matrix. For the first preconditioning matrix we form the Generalized Modified Extrapolated Successive Overrelaxation (GMESOR) method, whereas the second preconditioning matrix yields the Generalized Modified Preconditioned Simultaneous Displacement (GMPSD) method, which is an extrapolated form of the Symmetric Successive Overrelaxation method. We find sufficient conditions for each aforementioned iterative method to converge. In addition, we develop a geometric approach, for determining the optimum values of their parameters and corresponding spectral radii. It is shown that both iterative methods studied (GMESOR and GMPSD) attain the same rate of convergence. Numerical results confirm our theoretical expectations.