Backward error and condition of polynomial eigenvalue problems
type: Item
instance of: scholarly article
title: Backward error and condition of polynomial eigenvalue problems
zbMATH Open document ID: 0955.65027
DOI: 10.1016/S0024-3795(99)00063-4
author: Françoise Tisseur
published in: Linear Algebra and its Applications
publication date: 2001-02-14T00:00:00Z
review text: The author presents computable expressions for backward errors and condition numbers for the polynomial eigenvalue problem. Different ways to reformulate quadratic eigenvalue problems as generalized eigenvalue problems are considered. It is shown that backward stable algorithms for the generalized eigenvalue problem, e.g.~the QZ algorithm, can be backward unstable for the quadratic eigenvalue problem. For improving the backward error of the solutions obtained by means of the generalized eigenvalue problem formulation the possibility of using a scaling of the quadratic eigenvalue problem is studied. Furthermore, the sensitivity of a given eigenvalue to perturbations in the generalized eigenvalue problem formulations is analyzed. From this analysis one can conclude which formulation is more appropriate for the large and small eigenvalues. Numerical examples confirm this conclusions.
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 65H17
zbMATH DE Number: 1443456
zbMATH Keywords: polynomial eigenvalue problem
zbMATH Keywords: quadratic eigenvalue problem
zbMATH Keywords: generalized eigenvalue problem
zbMATH Keywords: backward error
zbMATH Keywords: condition number
zbMATH Keywords: algorithms
zbMATH Keywords: QZ algorithm
zbMATH Keywords: numerical examples
reviewed by: Michael Jung
describes a project that uses: testmatrix
describes a project that uses: mctoolbox
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1443456