  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.
