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Perturbation, computation and refinement of invariant pairs for matrix polynomials

by T. Betcke and D. Kressner

(Report number 2009-21)

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 \emph{polynomial} eigenvalue problems, for which the concept of invariant subspaces needs to be replaced by the concept of invariant pair. Little is 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.

Keywords:

@Techreport{BK09_407,
  author = {T. Betcke and D. Kressner},
  title = {Perturbation, computation and refinement of invariant pairs for matrix polynomials},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2009-21},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2009/2009-21.pdf },
  year = {2009}
}

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