Singular quadratic eigenvalue problems: Linearization and weak condition numbers

TL;DR

This paper investigates the conditioning of singular quadratic eigenvalue problems, demonstrating that proper linearizations and small perturbations can improve numerical stability and eigenvalue accuracy.

Contribution

It shows that suitable linearizations only marginally affect eigenvalue condition numbers and introduces a simple algorithm using random perturbations to identify well-conditioned eigenvalues.

Findings

Linearizations minimally increase $oldsymbol{ ext{ extdelta}}$-weak eigenvalue condition numbers.

Small random perturbations help detect and exclude spurious eigenvalues.

Eigenvalue condition numbers reliably indicate well-conditioned eigenvalues in singular problems.

Abstract

The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of $\delta$-weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases $\delta$-weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.