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Optimal a priori error bounds for the Rayleigh-Ritz method
Author(s):
Gerard
L. G.
Sleijpen;
Jasper
van den Eshof;
Paul
Smit.
Journal:
Math. Comp.
72
(2003),
677-684.
MSC (2000):
Primary 65F15;
Secondary 65F50
Posted:
May 1, 2002
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Abstract:
We derive error bounds for the Rayleigh-Ritz method for the approximation to extremal eigenpairs of a symmetric matrix. The bounds are expressed in terms of the eigenvalues of the matrix and the angle between the subspace and the eigenvector. We also present a sharp bound.
References:
-
- 1.
- Ernest R. Davidson, The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices, J. Comput. Phys. 17 (1975), 87-94. MR 52:2168
- 2.
- Zhongxiao Jia and G. W. Stewart, An analysis of the Rayleigh-Ritz method for approximating eigenspaces, Math. Comp. 70 (2001), no. 234, 637-647. MR 2001g:65040
- 3.
- Shmuel Kaniel, Estimates for some computational techniques in linear algebra, Math. Comp. 20 (1966), 369-378. MR 38 # 2934
- 4.
- Andrew V. Knyazev, New estimates for Ritz vectors, Math. Comp. 66 (1997), no. 219, 985-995. MR 97j:65090
- 5.
- Beresford N. Parlett, The symmetric eigenvalue problem, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998, Corrected reprint of the 1980 original. MR 99c:65072
- 6.
- Y. Saad, On the rates of convergence of the Lanczos and the block-Lanczos methods, SIAM J. Numer. Anal. 17 (1980), no. 5, 687-706. MR 82g:65022
- 7.
- Paul Smit, The approximation of an eigenvector by ritzvectors, Technical Report FEW 684, Center for Economic Research, University of Tilburg, Tilburg, The Netherlands, 1995.
- 8.
- -, Numerical analysis of eigenvalue algorithms based on subspace iterations, Ph.D. thesis, Center for Economic Research, Tilburg University, Tilburg, The Netherlands, July 1997.
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Additional Information:
Gerard
L. G.
Sleijpen
Affiliation:
Department of Mathematics, Utrecht University, P.O. Box 80.010, NL-3508 TA Utrecht, The Netherlands
Email:
sleijpen@math.uu.nl
Jasper
van den Eshof
Affiliation:
Department of Mathematics, Utrecht University, P.O. Box 80.010, NL-3508 TA Utrecht, The Netherlands
Email:
eshof@math.uu.nl
Paul
Smit
Affiliation:
Center for Economic Research, Tilburg University, Tilburg, The Netherlands
Address at time of publication:
IBM, Watsonweg 2, 1423 ND, Uithoorn, The Netherlands
Email:
p.smit@nl.ibm.com
DOI:
10.1090/S0025-5718-02-01435-7
PII:
S 0025-5718(02)01435-7
Keywords:
Symmetric matrices,
eigenvalue problem,
subspace projection,
Rayleigh-Ritz,
error bounds
Received by editor(s):
October 18, 2000
Received by editor(s) in revised form:
May 29, 2001
Posted:
May 1, 2002
Additional Notes:
The research of the second author was financially supported by the Dutch Scientific Organization (NWO), under project number 613.002.035
Copyright of article:
Copyright
2002,
American Mathematical Society
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