Approximation of the Pseudospectral Abscissa via Eigenvalue Perturbation Theory

Reliable and efficient computation of the pseudospectral abscissa in the large-scale setting is still not settled. Unlike the small-scale setting where there are globally convergent criss-cross algorithms, all algorithms in the large-scale setting proposed to date are at best locally convergent. We first describe how eigenvalue perturbation theory can be put in use to estimate the globally rightmost point in the $\epsilon$-pseudospectrum if $\epsilon$ is small. Our treatment addresses both general nonlinear eigenvalue problems, and the standard eigenvalue problem as a special case. For small $\epsilon$, the estimates by eigenvalue perturbation theory are quite accurate. In the standard eigenvalue case, we even derive a formula with an $\mathcal{O}({\epsilon }^3)$ error. For larger $\epsilon$, the estimates can be used to initialize the locally convergent algorithms. We also propose fixed-point iterations built on the the perturbation theory ideas for large $\epsilon$ that are suitable for the large-scale setting. The proposed fixed-point iterations initialized by using eigenvalue perturbation theory converge to the globally rightmost point in the pseudospectrum in a vast majority of the cases that we experiment with.