The Polynomial Eigenvalue Problem

Berhanu, Michael (2005) The Polynomial Eigenvalue Problem. Doctoral thesis, University of Manchester.

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In this thesis, we consider polynomial eigenvalue problems. We extend results on eigenvalue and eigenvector condition numbers of matrix polynomials to condition numbers with perturbations measured with a weighted Frobenius norm. We derive an explicit expression for the backward error of an approximate eigenpair of a matrix polynomial written in homogeneous form. We consider structured eigenvalue condition numbers for which perturbations have a certain structure such as symmetry, Hermitian or sparsity. We also obtain explicit and/or computable expressions for the structured backward error of an eigenpair. We present a robust implementation of the HZ (or HR) algorithm for symmetric generalized eigenvalue problems. This algorithm has the advantage of preserving pseudosymmetric tridiagonal forms. It has been criticized for its numerical instability. We propose an implementation of the HZ algorithm that allows stability in most cases and comparable results with other classical algorithms for ill conditioned problems. The HZ algorithm is based on the HR factorization, an extension of the QR factorization in which the H factor is hyperbolic. This yields us to the sensitivity analysis of hyperbolic factorizations.

Item Type: Thesis (Doctoral)
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Nick Higham
Date Deposited: 23 Aug 2006
Last Modified: 20 Oct 2017 14:12

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