On matrix nearness problems: Distance to delocalization

In this paper we introduce a new matrix nearness problem that is intended to generalize the distance to instability. Due to its applicability in analyzing the robustness of eigenvalues with respect to the arbitrary localization sets (domains) in the complex plane, we call it the distance to delocalization. For the open left half-plane or the unit disk, the distance to the nearest unstable matrix is obtained as a special case. Following the theoretical framework of Hermitian functions and the Lyapunov-type localization approach, we present new Newton-type algorithms for the distance to delocalization: first using an explicit computation of the desired singular values (eD2D), and then using an implicit computation (iD2D). For both algorithms, we introduce a special stabilization technique of the Newton steps and, for a certain class of the localization domains, we provide an additional globality test. Since our investigations are motivated by several practical applications, we illustrate our approach on some of them. Furthermore, in the special case when the distance to delocalization becomes the distance to the continuous time instability, we validate our algorithms against the state-of-the-art computational methods.