You are here: MIMS > EPrints
MIMS EPrints

2006.177: Finite precision behavior of stationary iteration for solving singular systems

2006.177: Nicholas J. Higham and Philip A. Knight (1993) Finite precision behavior of stationary iteration for solving singular systems. Linear Algebra and its Applications, 192. pp. 165-186. ISSN 0024-3795

Full text available as:

PDF - Archive staff only - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
1043 Kb

DOI: 10.1016/0024-3795(93)90242-G

Abstract

A stationary iterative method for solving a singular system Ax=b converges for any starting vector if limi→∞Gi exists, where G is the iteration matrix, and the solution to which it converges depends on the starting vector. We examine the behavior of stationary iteration in finite precision arithmetic. A pertubation bound for singular systems is derived and used to define forward stability of a numerical method. A rounding error analysis enables us to deduce conditions under which a stationary iterative method is forward stable or backward stable. The component of the forward error in the null space of A can grow linearly with the number of iterations, but it is innocuous as long as the iteration converges reasonably quickly. As special cases, we show that when A is symmetric positive semidefinite the Richardson iteration with optimal parameter is forward stable, and if A also has unit diagonal and property A, then the Gauss-Seidel method is both forward and backward stable. Two numerical examples are given to illustrate the analysis.

Item Type:Article
Subjects:MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2006.177
Deposited By:Miss Louise Stait
Deposited On:04 July 2006

Download Statistics: last 4 weeks
Repository Staff Only: edit this item