Hostname: page-component-7479d7b7d-c9gpj Total loading time: 0 Render date: 2024-07-12T10:17:11.399Z Has data issue: false hasContentIssue false
  • English
  • Français

A method of accelerating stationary iterative methods for solving linear systems

Published online by Cambridge University Press:  17 February 2009

G. K. Robinson
G. K. Robinson
Affiliation:
CSIRO Division of Mathematics and Statistics, Private Bag 10, Clayton, Vic. 3168, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The speed of convergence of stationary iterative techniques for solving simultaneous linear equations may be increased by using a method similar to conjugate gradients but which does not require the stationary iterative technique to be symmetrisable. The method of refinement is to find linear combinations of iterates from a stationary technique which minimise a quadratic form. This basic method may be used in several ways to construct refined versions of the simple technique. In particular, quadratic forms of much less than full rank may be used. It is suggested that the method is likely to be competitive with other techniques when the number of linear equations is very large and little is known about the properties of the system of equations. A refined version of the Gauss-Seidel technique was found to converge satisfactorily for two large systems of equations arising in the estimation of genetic merit of dairy cattle.

Type
Research Article
Information
The ANZIAM Journal , Volume 30 , Issue 1 , July 1988 , pp. 1 - 23
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Chatelin, F. and Miranker, W. L., “Acceleration by aggregation of successive approximation methods”, Linear Algebra Appl. 43 (1982) 17–47.CrossRefGoogle Scholar
[2]Eisenstat, S. C., Elman, H. C. and Schultz, M. H., “Variational iterative methods for nonsymmetric systems of linear equations”, SIAM J. Numer. Anal. 20 (1983) 345–357.CrossRefGoogle Scholar
[3]Everett, R. W., Quaas, R. L. and McClintock, A. C., “Dairy sire evaluation considering genetic merit of daughter's maternal grandsire”, J. Dairy Sci. 62 (1979) 1304–1313.CrossRefGoogle Scholar
[4]Hageman, L. A. and Young, D. M., Applied Iterative Methods (Academic Press, 1981).Google Scholar
[5]Henderson, C. R., “Use of all relatives in intraherd prediction of breeding values and producing abilities”, J. Dairy Sci. 58 (1975) 1910–1916.CrossRefGoogle Scholar
[6]Jennings, A., “Influence of the eigenvalue spectrum on the convergence rate of the conjugate gradient method”, J. Inst. Maths. Applies. 20 (1977) 61–72.CrossRefGoogle Scholar
[7]Martin, D. M. and Tee, G. J., “Iterative methods for linear equations with symmetric positive definite matrix”, Comput. J. 4 (1961) 242–254.CrossRefGoogle Scholar
[8]Varga, R. S., “A comparison of the successive over relaxation method and semi-iterative methods using Chebyshev polynomials”, J. Soc. Indus. Appl. Math. 5 (1957) 39–46.Google Scholar
[9]Young, D. M. and Jea, K. C., “Generalised conjugate gradient acceleration of non-symmetrisable iterative methods”, Linear Algebra Appl. 34 (1980) 159–194.Google Scholar