The Ehrlich-Aberth method for palindromic matrix polynomials represented in the Dickson basis

TL;DR

This paper introduces a specialized Ehrlich-Aberth based algorithm for efficiently computing eigenvalues of T-palindromic matrix polynomials using a structured linearization in the Dickson basis, enhancing robustness and speed.

Contribution

It presents a novel structured linearization in the Dickson basis and an adapted Ehrlich-Aberth method for palindromic matrix polynomials, improving computational efficiency and robustness.

Findings

Algorithm effectively computes eigenvalues with high accuracy.

Structured linearization exploits symmetry, reducing computational effort.

Numerical experiments demonstrate robustness and efficiency.

Abstract

An algorithm based on the Ehrlich-Aberth root-finding method is presented for the computation of the eigenvalues of a T-palindromic matrix polynomial. A structured linearization of the polynomial represented in the Dickson basis is introduced in order to exploit the symmetry of the roots by halving the total number of the required approximations. The rank structure properties of the linearization allow the design of a fast and numerically robust implementation of the root-finding iteration. Numerical experiments that confirm the effectiveness and the robustness of the approach are provided.