Solving polynomial eigenvalue problems by means of the Ehrlich-Aberth method

TL;DR

This paper presents an effective adaptation of the Ehrlich-Aberth method for solving polynomial eigenvalue problems, offering improved accuracy over traditional methods like QZ, and addresses structured matrix polynomials with special symmetries.

Contribution

It introduces a novel implementation of the Ehrlich-Aberth iteration tailored for polynomial eigenvalue problems, including structured matrices, with detailed computational strategies.

Findings

More accurate eigenvalue approximations than QZ method

Effective handling of structured matrix polynomials

Numerical experiments confirm the method's efficiency

Abstract

Given the $n\times n$ matrix polynomial $P(x)=\sum_{i=0}^kP_i x^i$, we consider the associated polynomial eigenvalue problem. This problem, viewed in terms of computing the roots of the scalar polynomial $\det P(x)$, is treated in polynomial form rather than in matrix form by means of the Ehrlich-Aberth iteration. The main computational issues are discussed, namely, the choice of the starting approximations needed to start the Ehrlich-Aberth iteration, the computation of the Newton correction, the halting criterion, and the treatment of eigenvalues at infinity. We arrive at an effective implementation which provides more accurate approximations to the eigenvalues with respect to the methods based on the QZ algorithm. The case of polynomials having special structures, like palindromic, Hamiltonian, symplectic, etc., where the eigenvalues have special symmetries in the complex plane, is considered. A general way to adapt the Ehrlich-Aberth iteration to structured matrix polynomial is introduced. Numerical experiments which confirm the effectiveness of this approach are reported.