Solving singular generalized eigenvalue problems. Part II: projection and augmentation

TL;DR

This paper introduces two novel methods for solving singular generalized eigenvalue problems, focusing on projection onto subspaces and augmented matrix pencils, to improve accuracy and efficiency over existing techniques.

Contribution

It develops two alternative approaches—projection and augmentation—for solving singular generalized eigenvalue problems, expanding on prior work with rank-completing additions.

Findings

Projection method is efficient for generic singular pencils.

Augmented pencil approach is suitable when linear systems can be solved efficiently.

Both methods improve accuracy and efficiency in solving singular eigenvalue problems.

Abstract

Generalized eigenvalue problems involving a singular pencil may be very challenging to solve, both with respect to accuracy and efficiency. While Part I presented a rank-completing addition to a singular pencil, we now develop two alternative methods. The first technique is based on a projection onto subspaces with dimension equal to the normal rank of the pencil while the second approach exploits an augmented matrix pencil. The projection approach seems to be the most attractive version for generic singular pencils because of its efficiency, while the augmented pencil approach may be suitable for applications where a linear system with the augmented pencil can be solved efficiently.