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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.
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Abstract

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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.

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

References

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