Backward error analysis of the shift-and-invert Arnoldi algorithm

Schröder, Christian and Taslaman, Leo (2014) Backward error analysis of the shift-and-invert Arnoldi algorithm. [MIMS Preprint]

WarningThere is a more recent version of this item available.
[img] PDF
MIMS.pdf

Download (475kB)

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.

Item Type: MIMS Preprint
Uncontrolled Keywords: backward error, shift-and-invert Arnoldi, inexact arithmetic, Krylov recurrence, eigenvalue, Hermitian matrix, stopping criterion, breakdown, modified Gram-Schmidt, QR factorization
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Leo Taslaman
Date Deposited: 23 Oct 2014
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2190

Available Versions of this Item

Actions (login required)

View Item View Item