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

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

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.

**1.**R. Bhatia, L. Elsner, and G. Krause,*Bounds for the variation of the roots of a polynomial and the eigenvalues of a matrix*, Linear Algebra and Its Applications**142**(1990), 195-209. MR**92i:12001****2.**L. Elsner,*An optimal bound for the spectral variation of two matrices*, Linear Algebra and Its Applications**71**(1985), 77-80. MR**87c:15035****3.**G. H. Golub and C. F. Van Loan,*Matrix computations*, second ed., Johns Hopkins University Press, Baltimore, MD, 1989. MR**90d:65055****4.**I. C. F. Ipsen,*Absolute and relative perturbation bounds for invariant subspaces of matrices*, Technical Report TR97-35, Center for Research in Scientific Computation, Mathematics Department, North Carolina State Unversity, 1998.**5.**Z. Jia,*Some numerical methods for large unsymmetric eigenproblems*, Ph.D. thesis, University of Bielefeld, 1994.**6.**-,*The convergence of generalized Lanczos methods for large unsymmetric eigenproblems*, SIAM Journal on Matrix Analysis and Applications**16**(1995), 843-862. MR**96d:65062****7.**-,*Refined iterative algorithm based on Arnoldi's process for large unsymmetric eigenproblems*, Linear Algebra and Its Applications**259**(1997), 1-23. MR**98c:65060****8.**-,*Generalized block Lanczos methods for large unsymmetric eigenproblems*, Numerische Mathematik**80**(1998), 171-189. MR**95f:65059****9.**-,*A refined iterative algorithm based on the block Arnoldi algorithm*, Linear Algebra and Its Applications**270**(1998), 171-189. MR**98m:65055****10.**-,*Polynomial characterizations of the approximate eigenvectors by the refined Arnoldi method and an implicitly restarted refined Arnoldi algorithm*, Linear Algebra and Its Applications**287**(1999), 191-214. MR**99j:65046****11.**-,*A refined subspace iteration algorithm for large sparse eigenproblems*, To appear in*Applied Numerical Mathemtics.*, 1999.**12.**Y. Saad,*Numerical methods for large eigenvalue problems: Theory and algorithms*, John Wiley, New York, 1992. MR**93h:65052****13.**G. L. G. Sleijpen and H. A. Van der Vorst,*A Jacobi-Davidson iteration method for linear eigenvalue problems*, SIAM Journal on Matrix Analysis and Applications**17**(1996), 401-425. MR**96m:65042****14.**G. W. Stewart and J.-G. Sun,*Matrix perturbation theory*, Academic Press, New York, 1990. MR**92a:65017**

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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:
https://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