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Mathematics of Computation

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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: This paper concerns the Rayleigh–Ritz method for computing an approximation to an eigenspace $\mathcal {X}$ of a general matrix $A$ from a subspace $\mathcal {W}$ that contains an approximation to $\mathcal {X}$. The method produces a pair $(N, \tilde X)$ that purports to approximate a pair $(L, X)$, where $X$ is a basis for $\mathcal {X}$ and $AX = XL$. In this paper we consider the convergence of $(N, \tilde X)$ as the sine $\epsilon$ of the angle between $\mathcal {X}$ and $\mathcal {W}$ approaches zero. It is shown that under a natural hypothesis — called the uniform separation condition — the Ritz pairs $(N, \tilde X)$ converge to the eigenpair $(L, X)$. 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 $A$ 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

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

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