Solving singular generalized eigenvalue problems. Part III: structure preservation

TL;DR

This paper introduces structure-preserving methods for solving singular Hermitian eigenvalue problems, extending previous techniques to maintain symmetry and sign characteristics, and applies these methods to bivariate polynomial systems.

Contribution

It develops Hermitian perturbation-based structure-preserving algorithms for singular pencils, addressing previous limitations in symmetric cases and preserving key spectral properties.

Findings

Methods successfully preserve structure and sign characteristic.

Algorithms effectively solve systems of bivariate polynomials.

Approach extends previous techniques to Hermitian and symmetric pencils.

Abstract

In Parts I and II of this series of papers, three new methods for the computation of eigenvalues of singular pencils were developed: rank-completing perturbations, rank-projections, and augmentation. It was observed that a straightforward structure-preserving adaption for symmetric pencils was not possible and it was left as an open question how to address this challenge. In this Part III, it is shown how the observed issue can be circumvented by using Hermitian perturbations. This leads to structure-preserving analogues of the three techniques from Parts I and II for Hermitian pencils (including real symmetric pencils) as well as for related structures. It is an important feature of these methods that the sign characteristic of the given pencil is preserved. As an application, it is shown that the resulting methods can be used to solve systems of bivariate polynomials.