An Iterative Method for Computing the Pseudospectral Abscissa for a Class of Nonlinear Eigenvalue Problems

Abstract

We consider the following class of nonlinear eigenvalue problems: $\left(\sum_{i=1}^m A_i p_i(\lambda)\right) v= 0,$ where $A_1, \ldots,A_m$ are given $n \times n$ matrices and the functions $p_1, \ldots, p_m$ are assumed to be entire. This does not only include polynomial eigenvalue problems but also eigenvalue problems arising from systems of delay differential equations. Our aim is to compute the $\epsilon$-pseudospectral abscissa, i.e., the supremum of the real parts of the points in the $\epsilon$-pseudospectrum, which is the complex set obtained by joining all solutions of the eigenvalue problem under perturbations $\{ \delta A_i \}_{i=1}^{m}$, of norm at most $\epsilon$, of the matrices $\{ A_i \}_{i=1}^{m}$. Under mild assumptions, guaranteeing the existence of a globally rightmost point of the $\epsilon$-pseudospectrum, we prove that it is sufficient to restrict the analysis to rank-one perturbations of the form $\delta A_i = \beta_i u v^*$, where $u \in \mathbb{C}^n$ and $v \in \mathbb{C}^n$ wit...