Optimal scaling of generalized and polynomial eigenvalue problems

Betcke, T. (2007) Optimal scaling of generalized and polynomial eigenvalue problems. [MIMS Preprint]

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Scaling is a commonly used technique for standard eigenvalue problems to improve the sensitivity of the eigenvalues. In this paper we investigate scaling for generalized and polynomial eigenvalue problems (PEPs) of arbitrary degree. It is shown that an optimal diagonal scaling of a PEP with respect to an eigenvalue can be described by the ratio of its normwise and componentwise condition number. Furthermore, the effect of linearization on optimally scaled polynomials is investigated. We introduce a generalization of the diagonal scaling by Lemonnier and Van Dooren to PEPs that is especially effective if some information about the magnitude of the wanted eigenvalues is available and also discuss variable transformations of the type λ = αμ for PEPs of arbitrary degree.

Item Type: MIMS Preprint
Additional Information: submitted to SIAM J. Matrix Anal. Appl. (revised version)
Uncontrolled Keywords: polynomial eigenvalue problem, balancing, scaling, condition number, backward error
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Dr. Timo Betcke
Date Deposited: 10 Apr 2008
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/1075

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