Solving singular generalized eigenvalue problems by a rank-completing perturbation

TL;DR

This paper introduces a simple, fast, and robust perturbation-based method for solving singular generalized eigenvalue problems, offering an efficient alternative to existing staircase methods.

Contribution

The paper presents a novel perturbation technique for accurately computing eigenvalues of singular pencils, improving efficiency and robustness over traditional methods.

Findings

Method is faster and more robust than existing approaches.

Applicable to a wide range of singular eigenvalue problems.

Provides an efficient alternative to staircase methods.

Abstract

Generalized eigenvalue problems involving a singular pencil are very challenging to solve, both with respect to accuracy and efficiency. The existing package Guptri is very elegant but may sometimes be time-demanding, even for small and medium-sized matrices. We propose a simple method to compute the eigenvalues of singular pencils, based on one perturbation of the original problem of a certain specific rank. For many problems, the method is both fast and robust. This approach may be seen as a welcome alternative to staircase methods.