Hostname: page-component-cb9f654ff-nr592 Total loading time: 0 Render date: 2025-08-23T09:32:44.103Z Has data issue: false hasContentIssue false
  • English
  • Français

Rank-1 perturbations and the Lanczos method, inverse iteration, and Krylov subspaces

Published online by Cambridge University Press:  17 February 2009

Christopher T. Lenard
Christopher T. Lenard
Affiliation:
Department of Mathematics, LaTrobe University, Bendigo, PO Box 199, Bendigo VIC 3550, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The heart of the Lanczos algorithm is the systematic generation of orthonormal bases of invariant subspaces of a perturbed matrix. The perturbations involved are special since they are always rank-1 and are the smallest possible in certain senses. These minimal perturbation properties are extended here to more general cases.

Rank-1 perturbations are also shown to be closely connected to inverse iteration, and thus provide a novel explanation of the global convergence phenomenon of Rayleigh quotient iteration.

Finally, we show that the restriction to a Krylov subspace of a matrix differs from the restriction of its inverse by a rank-1 matrix.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 36 , Issue 4 , April 1995 , pp. 381 - 388
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Golub, G. H. and Loan, C. F. Van, Matrix computations, 2 ed. (Johns Hopkins University Press, Baltimore, 1989).Google Scholar
[2]Lenard, C. T., “Rank-1 homotopy methods for eigenproblems”, submitted for publication.Google Scholar
[3]Parlett, B. N., “The Rayleigh quotient iteration and some generalizations for non-normal matrices”, Math. Comp. 13 (1974) 679–693.CrossRefGoogle Scholar
[4]Parlett, B. N., The symmetric eigenvalue problem (Prentice-Hall, Englewood Cliffs, New Jersey, 1980).Google Scholar
[5]Sloan, I. H., “Iterated Galerkin method for eigenvalue problems”, SIAM J. Numer. Anal. 13 (1976) 753–760.CrossRefGoogle Scholar