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Main Title: Backward error analysis of the shift-and-invert Arnoldi algorithm
Author(s): Schröder, Christian
Taslaman, Leo
Type: Article
Language Code: en
Abstract: We perform a backward error analysis of the inexact shift-and-invert Arnoldi algorithm. We consider inexactness in the solution of the arising linear systems, as well as in the orthonormalization steps, and take the non-orthonormality of the computed Krylov basis into account. We show that the computed basis and Hessenberg matrix satisfy an exact shift-and-invert Krylov relation for a perturbed matrix, and we give bounds for the perturbation. We show that the shift-and-invert Arnoldi algorithm is backward stable if the condition number of the small Hessenberg matrix is not too large. This condition is then relaxed using implicit restarts. Moreover, we give notes on the Hermitian case, considering Hermitian backward errors, and finally, we use our analysis to derive a sensible breakdown condition.
Issue Date: 2015
Date Available: 28-Aug-2017
DDC Class: 511 Allgemeine mathematische Prinzipien
510 Mathematik
Subject(s): MSC 65F25
MSC 65F50
MSC 65G50
Journal Title: Numerische Mathematik
Publisher: Springer
Publisher Place: Berlin, Heidelberg
Volume: 133
Issue: 4
Publisher DOI: 10.1007/s00211-015-0759-9
Page Start: 819
Page End: 843
EISSN: 0945-3245
ISSN: 0029-599X
Appears in Collections:FG Numerische Mathematik » Publications

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