Chapter

Short Recurrences for Generating Orthogonal Krylov Subspace Bases

Jörg Liesen and Zdenek Strakos

in Krylov Subspace Methods

Published in print October 2012 | ISBN: 9780199655410
Published online January 2013 | e-ISBN: 9780191744174 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199655410.003.0004

Series: Numerical Mathematics and Scientific Computation

Short Recurrences for Generating Orthogonal Krylov Subspace Bases

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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

Chapter.  35102 words.  Illustrated.

Subjects: Applied Mathematics

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