Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information


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.

References [Enhancements On Off] (What's this?)

  • 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 Appl. 142 (1990), 195–209. MR 1077985, 10.1016/0024-3795(90)90267-G
  • 2. L. Elsner, An optimal bound for the spectral variation of two matrices, Linear Algebra Appl. 71 (1985), 77–80. MR 813034, 10.1016/0024-3795(85)90236-8
  • 3. Gene H. Golub and Charles F. Van Loan, Matrix computations, 2nd ed., Johns Hopkins Series in the Mathematical Sciences, vol. 3, Johns Hopkins University Press, Baltimore, MD, 1989. MR 1002570
  • 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. Zhong Xiao Jia, The convergence of generalized Lanczos methods for large unsymmetric eigenproblems, SIAM J. Matrix Anal. Appl. 16 (1995), no. 3, 843–862. MR 1337649, 10.1137/S0895479893246753
  • 7. Zhongxiao Jia, Refined iterative algorithms based on Arnoldi’s process for large unsymmetric eigenproblems, Linear Algebra Appl. 259 (1997), 1–23. MR 1450527, 10.1016/S0024-3795(96)00238-8
  • 8. J. Saranen and L. Schroderus, The modified quadrature method for classical pseudodifferential equations of negative order on smooth closed curves, Proceedings of the Fifth International Congress on Computational and Applied Mathematics (Leuven, 1992), 1994, pp. 485–495. MR 1284284, 10.1016/0377-0427(94)90322-0
  • 9. Zhongxiao Jia, A refined iterative algorithm based on the block Arnoldi process for large unsymmetric eigenproblems, Linear Algebra Appl. 270 (1998), 171–189. MR 1484080
  • 10. Zhongxiao Jia, Polynomial characterizations of the approximate eigenvectors by the refined Arnoldi method and an implicitly restarted refined Arnoldi algorithm, Linear Algebra Appl. 287 (1999), no. 1-3, 191–214. Special issue celebrating the 60th birthday of Ludwig Elsner. MR 1662868, 10.1016/S0024-3795(98)10197-0
  • 11. -, A refined subspace iteration algorithm for large sparse eigenproblems, To appear in Applied Numerical Mathemtics., 1999.
  • 12. Youcef Saad, Numerical methods for large eigenvalue problems, Algorithms and Architectures for Advanced Scientific Computing, Manchester University Press, Manchester; Halsted Press [John Wiley & Sons, Inc.], New York, 1992. MR 1177405
  • 13. Gerard L. G. Sleijpen and Henk A. Van der Vorst, A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl. 17 (1996), no. 2, 401–425. MR 1384515, 10.1137/S0895479894270427
  • 14. G. W. Stewart and Ji Guang Sun, Matrix perturbation theory, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1990. MR 1061154

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 15A18, 65F15, 65F50

Retrieve articles in all journals with MSC (2000): 15A18, 65F15, 65F50

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