Solving singular generalized eigenvalue problems. Part III: structure preservation

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.