An analysis of the Rayleigh-Ritz method for approximating eigenspaces

Authors:
Zhongxiao Jia and G. W. Stewart

Journal:
Math. Comp. **70** (2001), 637-647

MSC (2000):
Primary 15A18, 65F15, 65F50

Published electronically:
February 18, 2000

MathSciNet review:
1697647

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Abstract | References | Similar Articles | Additional Information

This paper concerns the Rayleigh-Ritz method for computing an approximation to an eigenspace of a general matrix from a subspace that contains an approximation to . The method produces a pair that purports to approximate a pair , where is a basis for and . In this paper we consider the convergence of as the sine of the angle between and approaches zero. It is shown that under a natural hypothesis--called the uniform separation condition--the Ritz pairs converge to the eigenpair . When one is concerned with eigenvalues and eigenvectors, one can compute certain refined Ritz vectors whose convergence is guaranteed, even when the uniform separation condition is not satisfied. An attractive feature of the analysis is that it does not assume that has distinct eigenvalues or is diagonalizable.

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Additional Information

**Zhongxiao Jia**

Affiliation:
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, P.R. China

Email:
zxjia@dlut.edu.cn

**G. W. Stewart**

Affiliation:
Department of Computer Science, Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742, USA

Email:
stewart@cs.umd.edu

DOI:
http://dx.doi.org/10.1090/S0025-5718-00-01208-4

Received by editor(s):
April 9, 1998

Received by editor(s) in revised form:
May 5, 1999

Published electronically:
February 18, 2000

Additional Notes:
The first author’s work was supported by the China State Major Key Project for Basic Researches, the National Natural Science Foundation of China, the Foundation for Excellent Young Scholars of the Ministry of Education and the Doctoral Point Program of the Ministry of Education, China.

The second author’s work was supported by the National Science Foundation under Grant No. 970909-8562.

Article copyright:
© Copyright 2000
American Mathematical Society