On the computation of eigenvalues, spectral bounds, and Hessenberg form for matrix polynomials

Abstract

In this dissertation we focus on root-finding methods, such as Laguerre's method, for solving the polynomial eigenvalue problem. Serious consideration is given to the initial conditions and stopping criteria. Cost efficient and accurate strategies for computing eigenvectors, backward error, and condition estimates are given. Applications for both Hessenberg and tridiagonal structure are provided, and it shown that significant computational savings can be made from both structures. Surprising results concerning the spectral bounds for unitary matrix polynomials are presented. In addition, a constructive proof is provided for the result that every square matrix polynomial can be reduced to an upper Hessenberg matrix whose entries are rational functions and in special cases polynomials. The determinant of the matrix polynomial is preserved under this transformation, and sufficient conditions are provided for which the Smith form is preserved.