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]

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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:	29 Oct 2014
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/2192

Available Versions of this Item

• Backward error analysis of the shift-and-invert Arnoldi algorithm. (deposited 23 Oct 2014)
  • Backward error analysis of the shift-and-invert Arnoldi algorithm. (deposited 29 Oct 2014) [Currently Displayed]

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