Structured eigenvalue backward errors of matrix pencils and polynomials with Hermitian and related structures
type: Item
instance of: scholarly article
zbMATH Open document ID: 1304.15016
DOI: 10.1137/130925621
author: Shreemayee Bora
author: Michael Karow
author: Christian Mehl
author: Punit Sharma
published in: SIAM Journal on Matrix Analysis and Applications
publication date: 2014-10-15T00:00:00Z
full work available at URL: https://depositonce.tu-berlin.de//handle/11303/7279
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15A22
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 15A54
zbMATH DE Number: 6356180
zbMATH Keywords: eigenvalue
zbMATH Keywords: perturbations
zbMATH Keywords: Hermitian matrix pencil
zbMATH Keywords: Hermitian matrix polynomial
zbMATH Keywords: eigenvalue backward error
zbMATH Keywords: structured mapping problem
MaRDI profile type: Publication
OpenAlex ID: W1964067206
title: Structured eigenvalue backward errors of matrix pencils and polynomials with Hermitian and related structures
review text: Given a regular structured matrix polynomial \(P(z)= z^kA_k+\cdots +zA_1+A_0\) with \(A_0, \dots, A_k\in {\mathbb C}^{n\times n}\). The authors answer the following question: If \(\lambda \in {\mathbb C}\), what is the smallest perturbation \((\Delta_0, \dots, \Delta_k)\) from some perturbation set \(S\subseteq ({\mathbb C}^{n\times n})^{k+1}\) such that \(\lambda\) is an eigenvalue of \(P(z)=z^k(A_k-\Delta_k)+\cdots +z(A_1-\Delta_1)+(A_0-\Delta_0)\)? This question is answered for matrix pencils as well as for polynomials which are Hermitian, skew-Hermitian, \(*\)-even and \(*\)-odd. The authors carry out numerical experiments from which they are able to conclude that, in many cases there is a significant difference between the backward errors with respect to perturbations that preserve structure and those with respect to arbitrary perturbations.
reviewed by: David Ssevviiri
description: scientific article; zbMATH DE number 6356180