The Polynomial Eigenvalue Problem
Michaël Berhanu · MIMS EPrints (University of Southampton) · 2005
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.