# Algorithms for Matrix Polynomials and Structured Matrix Problems

Munro, Christopher J. (2011) Algorithms for Matrix Polynomials and Structured Matrix Problems. Doctoral thesis, Manchester Institute for Mathematical Sciences, The University of Manchester.

In this thesis we focus on algorithms for matrix polynomials and structured matrix problems. We begin by presenting a general purpose eigensolver for dense quadratic eigenvalue problems, which incorporates recent contributions on the numerical solution of polynomial eigenvalue problems, namely a scaling of the eigenvalue parameter prior to the computation, and a choice of linearization with favourable conditioning and backward stability properties. Our algorithm includes a preprocessing step that reveals the zero and infinite eigenvalues contributed by singular leading and trailing matrix coefficients and deflates them. Numerical experiments are presented, comparing the performance of this algorithm on a collection of test problems, in terms of accuracy and stability. We then describe structure preserving transformations for quadratic matrix polynomials. Given a pair of distinct eigenvalues s $(\l_1,\l_2)$ of an n-by-n quadratic matrix polynomial $Q(\l)=\l^2 A_2+\l A_1 +A_0$ with a nonsingular leading coefficient and their corresponding eigenvectors, we show how to transform $Q(\l)$ into a quadratic of the form $\mmtwobytwomatrix{\Qd(\l)}{0}{0}{q(\l)}$ having the same eigenvalues as $Q(\l)$, with $\Qd(\l)$ an $(n-1)\times (n-1)$ quadratic matrix polynomial and $q(\l)$ a scalar quadratic polynomial with roots $\l_1$ and $\l_2$.