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Main Title: On Matrix Nearness Problems: Distance to Delocalization
Author(s): Kostic, Vladimir
Miedlar, Agnieszka
Stolwijk, Jeroen
Type: Research Paper
URI: https://depositonce.tu-berlin.de/handle/11303/15822
http://dx.doi.org/10.14279/depositonce-14595
License: http://rightsstatements.org/vocab/InC/1.0/
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.
Subject(s): stability
matrix nearness problems
distance to instability
distance to stability
spectral abscissa
stability radius
$\eps$-pseudospectra
Lyapunov stability test
Hermitian functions
Newton's method
Issue Date: 2-Apr-2014
Date Available: 17-Dec-2021
Language Code: en
DDC Class: 510 Mathematik
MSC 2000: 65F15 Eigenvalues, eigenvectors
15A18 Eigenvalues, singular values, and eigenvectors
15A22 Matrix pencils
Series: Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin
Series Number: 2014, 07
ISSN: 2197-8085
TU Affiliation(s): Fak. 2 Mathematik und Naturwissenschaften » Inst. Mathematik
Appears in Collections:Technische Universität Berlin » Publications

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