Numerical methods for eigenvalues of singular polynomial eigenvalue problems

TL;DR

This paper generalizes three numerical methods for singular matrix pencils to singular polynomial eigenvalue problems, enabling the separation of true eigenvalues from fake ones using eigenvector information.

Contribution

It extends existing methods based on rank-completing perturbation, rank-projection, and augmentation to handle singular polynomial eigenvalue problems.

Findings

All three approaches can be adapted to singular polynomial problems.

Eigenvalues can be separated into true and fake using eigenvector data.

Applications include bivariate polynomial systems and ZGV points.

Abstract

Recently, three numerical methods for the computation of eigenvalues of singular matrix pencils, based on a rank-completing perturbation, a rank-projection, or an augmentation were developed. We show that all three approaches can be generalized to treat singular polynomial eigenvalue problems. The common denominator of all three approaches is a transformation of a singular into a regular matrix polynomial whose eigenvalues are a disjoint union of the eigenvalues of the singular polynomial, called true eigenvalues, and additional fake eigenvalues. The true eigenvalues can then be separated from the fake eigenvalues using information on the corresponding left and right eigenvectors. We illustrate the approaches on several interesting applications, including bivariate polynomial systems and ZGV points.