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Perturbation, extraction and refinement of invariant pairs for matrix polynomials

Betcke, T; Kressner, D; (2011) Perturbation, extraction and refinement of invariant pairs for matrix polynomials. Linear Algebra and its Applications , 435 (3) pp. 514-536. 10.1016/j.laa.2010.06.029. Green open access

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Abstract

Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar benefits can be expected for polynomial eigenvalue problems, for which the concept of an invariant subspace needs to be replaced by the concept of an invariant pair. Little has been known so far about numerical aspects of such invariant pairs. The aim of this paper is to fill this gap. The behavior of invariant pairs under perturbations of the matrix polynomial is studied and a first-order perturbation expansion is given. From a computational point of view, we investigate how to best extract invariant pairs from a linearization of the matrix polynomial. Moreover, we describe efficient refinement procedures directly based on the polynomial formulation. Numerical experiments with matrix polynomials from a number of applications demonstrate the effectiveness of our extraction and refinement procedures. (C) 2010 Elsevier Inc. All rights reserved.

Type: Article
Title: Perturbation, extraction and refinement of invariant pairs for matrix polynomials
Open access status: An open access version is available from UCL Discovery
DOI: 10.1016/j.laa.2010.06.029
Publisher version: http://dx.doi.org/10.1016/j.laa.2010.06.029
Language: English
Keywords: Polynomial eigenvalue problem, Invariant pairs, Numerical algorithm, Perturbation theory, QUADRATIC EIGENVALUE PROBLEMS, NUMERICAL-SOLUTION, SUBSPACES, LINEARIZATIONS, ALGORITHM, BOUNDS, ERROR
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/1056733
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