×

A note on backward perturbations for the Hermitian eigenvalue problem. (English) Zbl 0841.65023

The paper studies two kinds of optimal Hermitian backward perturbations for the Hermitian eigenvalue problem which are obtained from different orthogonal decompositions of computed eigenvectors. It is shown that small residuals and almost orthogonality of the computed eigenvectors does not necessarily imply the smallness of all those optimal perturbations. This contradicts a result by S. Chandrasekharan and I. C. F. Ipsen [Numer. Math. 68, No. 2, 215-223 (1994; Zbl 0807.65034)] for the real symmetric eigenvalue problem.

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices

Citations:

Zbl 0807.65034

Software:

IPSEN
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] S. Chandrasekaran and I. C. F. Ipsen,Backward errors for eigenvalue and singular value decompositions, Numer. Math., 68 (1994), pp. 215–223. · Zbl 0807.65034 · doi:10.1007/s002110050057
[2] C. -H. Chen and J. -G. Sun,Perturbation bounds for the polar factors, J. Comput. Math., 7 (1989), pp. 397–401. · Zbl 0714.15009
[3] G. H. Golub and C. Van Loan,Matrix Computations, 2nd Edition, The Johns Hopkins University Press, Baltimore, Maryland, 1989. · Zbl 0733.65016
[4] N. J. Higham,Computing the polar decomposition - with applications, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 1160–1174. · Zbl 0607.65014 · doi:10.1137/0907079
[5] W. M. Kahan,Inclusion theorems for clusters of eigenvalues of Hermitian matrices, Technical Report, Computer Science Department, University of Toronto, 1967. · Zbl 0189.48001
[6] R. -C. Li,A bound on perturbations for polar decompositions, to appear in SIAM J. Matrix Anal. Appl.
[7] G. W. Stewart and J. -G. Sun,Matrix Perturbation Theory, Academic Press, New York, 1990. · Zbl 0706.65013
[8] J. -G. Sun,Perturbation bounds for the Cholesky and QR factorizations, BIT, 31 (1991), pp. 341–352. · Zbl 0728.65032 · doi:10.1007/BF01931293
[9] J. -G. Sun,Backward perturbation analysis of certain characteristic subspaces, Numer. Math., 65 (1993), pp. 357–382. · Zbl 0791.65023 · doi:10.1007/BF01385757
[10] J. -G. Sun,A posteriori error bounds for eigenvalues and singular values, Report UMINF-94.05, ISSN-0348-0542, Department of Computing Science, Umeå University, Sweden, 1994.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.