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Krylov Subspace MethodsPrinciples and Analysis$
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Jörg Liesen and Zdenek Strakos

Print publication date: 2012

Print ISBN-13: 9780199655410

Published to Oxford Scholarship Online: January 2013

DOI: 10.1093/acprof:oso/9780199655410.001.0001

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Short Recurrences for Generating Orthogonal Krylov Subspace Bases

Short Recurrences for Generating Orthogonal Krylov Subspace Bases

(p.168) 4 Short Recurrences for Generating Orthogonal Krylov Subspace Bases
Krylov Subspace Methods

Jörg Liesen

Zdenek Strakos

Oxford University Press

This chapter links Krylov subspace methods to classical topics of linear algebra. The main goal is to explain when a Krylov sequence can be orthogonalised with an optimal (Arnoldi-type) short recurrence. This question was posed by Golub in 1981 and answered by the Faber–Manteuffel theorem in 1984. The chapter gives a new complete proof of this theorem that has not been published elsewhere. It is based on the cyclic decomposition of a vector space with respect to a given linear operator. The theorem motivates the theoretically and practically important distinction made between Hermitian and non-Hermitian problems in the area of Krylov subspace methods. The matrix-version of the theorem works with the so-called B-normal(s) matrices, and this property is linked for a general matrix with the number of its distinct eigenvalues. The chapter also reviews other types of recurrences and it ends with brief remarks on integral representations of invariant subspaces.

Keywords:   cyclic invariant subspaces, Jordan canonical form, length of Krylov sequences, optimal short recurrences, Faber–Manteuffel theorem, b-normal(s) property, harmonic polynomials, isometric Arnoldi algorithm, generalised Lanczos algorithm, cauchy integral representation

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