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2007.120: Optimal scaling of generalized and polynomial eigenvalue problems

2007.120: T. Betcke (2007) Optimal scaling of generalized and polynomial eigenvalue problems.

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Abstract

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.

Uncontrolled Keywords:polynomial eigenvalue problem, balancing, scaling, condition number, backward error
Subjects:MSC 2000 > 65 Numerical analysis
MIMS number:2007.120
Deposited By:Dr. Timo Betcke
Deposited On:08 October 2007

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