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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

Chapter:
(p.168) 4 Short Recurrences for Generating Orthogonal Krylov Subspace Bases
Source:
Krylov Subspace Methods
Author(s):

Jörg Liesen

Zdenek Strakos

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199655410.003.0004

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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