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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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An analysis of the Rayleigh–Ritz method for approximating eigenspaces
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by Zhongxiao Jia and G. W. Stewart PDF
Math. Comp. 70 (2001), 637-647 Request permission

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
  • 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
  • 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.
  • © Copyright 2000 American Mathematical Society
  • Journal: Math. Comp. 70 (2001), 637-647
  • MSC (2000): Primary 15A18, 65F15, 65F50
  • DOI: https://doi.org/10.1090/S0025-5718-00-01208-4
  • MathSciNet review: 1697647