direkt zum Inhalt springen

direkt zum Hauptnavigationsmenü

Sie sind hier

TU Berlin

Inhalt des Dokuments

Preprint 07-2014

On Matrix Nearness Problems: Distance to Delocalization

Source file is available as :   Portable Document Format (PDF)

Author(s) : Vladimir Kostic , Agnieszka Miedlar , Jeroen Stolwijk

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 07-2014

MSC 2000

65F15 Eigenvalues, eigenvectors
15A18 Eigenvalues, singular values, and eigenvectors

Abstract :
This paper introduces two new matrix nearness problems that are intended to generalize the distance to instability and the distance to stability. They are named the distance to delocalization and the distance to localization due to their applicability in analyzing the robustness of eigenvalues with respect to arbitrary localization sets (domains) in the complex plane. For the open left-half plane or the unit circle, the distance to the nearest unstable/stable matrix is obtained as a special case. Then, following the theoretical framework of Hermitian functions and the Lyapunov-type localization approach, we present a new Newton-type algorithm for the distance to delocalization (D2D) and study its implementations using both an explicit and an implicit computation of the desired singular values. Since our investigations are motivated by several practical applications, we will illustrate our approach on some of them. Furthermore, in the special case when the distance to delocalization becomes the distance to instability, we will validate our algorithms against the state of the art computational method.

Keywords : stability, matrix nearness problems, distance to instability, distance to stability, spectral abscissa, stability radius, $\eps$-pseudospectra, Lyapunov stability test, Hermitian functions, Newton's method

Zusatzinformationen / Extras


Schnellnavigation zur Seite über Nummerneingabe