## 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

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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 |
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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 |

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