A filtering method for the interval eigenvalue problem
type: Item
instance of: scholarly article
title: A filtering method for the interval eigenvalue problem
zbMATH Open document ID: 1221.65095
author: Milan Hladík
author: David Daney
author: Elias P. Tsigaridas
published in: Applied Mathematics and Computation
publication date: 2011-03-08T00:00:00Z
full work available at URL: RR-7057.pdf
review text: The real eigenvalue set \(\Lambda({\mathbf A})\) of an interval matrix \({\mathbf A}= [\underline A,\overline A]\) is defined by \(\Lambda({\mathbf A})= \{\lambda\in\mathbb{R}\mid Ax=\lambda x\), \(x\neq 0\), \(A\in{\mathbf A}\}\). A filtering method for \(\Lambda({\mathbf A})\) is a method which iteratively cuts off some parts from a given interval \({\mathbf a}\) that do not include any eigenvalue. Finally it delivers an interval \({\mathbf b}\subseteq{\mathbf a}\) such that \(({\mathbf a}\setminus{\mathbf b})\cap\Lambda({\mathbf A})=\emptyset\). The authors present such a filtering method based on a sufficient regularity condition for a shifted matrix of \({\mathbf A}\). They apply it also to symmetric interval matrices \({\mathbf A}^S= \{A\in{\mathbf A}\mid A= A^T\}\) in order to get enclosures of the corresponding eigenvalue set \(\Lambda({\mathbf A}^S)\subseteq\Lambda({\mathbf A})\). By three numerical examples they demonstrate the efficiency of their method.
reviewed by: Günter Mayer
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 65G30
Mathematics Subject Classification ID: 15A42
zbMATH DE Number: 5862460
zbMATH Keywords: interval matrix
zbMATH Keywords: symmetric matrix
zbMATH Keywords: interval analysis
zbMATH Keywords: eigenvalue
zbMATH Keywords: eigenvalue bounds
zbMATH Keywords: eigenvalue set of an interval matrix
zbMATH Keywords: filtering method
zbMATH Keywords: numerical examples
describes a project that uses: PROFIL/BIAS
describes a project that uses: GLPK
describes a project that uses: CLAPACK
MaRDI profile type: Publication
OpenAlex ID: W2138446884
cites work: An approximate method for the standard interval eigenvalue problem of real non-symmetric interval matrices
cites work: Interval analysis of vibrating systems
cites work: Principal component analysis on interval data
cites work: Independent component analysis, a new concept?
cites work: Q5689624
cites work: Matrix Analysis
cites work: The Interval Eigenvalue Problem
cites work: On the range of eigenvalues of an interval matrix
cites work: Interval Matrices: Singularity and Real Eigenvalues
cites work: Computing bounds to real eigenvalues of real-interval matrices
cites work: An algorithm for addressing the real interval eigenvalue problem
cites work: An evolution strategy method for computing eigenvalue bounds of interval matrices
cites work: Computing eigenvalue bounds of structures with uncertain‐but‐non‐random parameters by a method based on perturbation theory
cites work: Bounds on Real Eigenvalues and Singular Values of Interval Matrices
cites work: Outer interval solution of the eigenvalue problem under general form parametric dependencies
cites work: Q4058008
cites work: Q3730926
cites work: Sufficient Conditions for Regularity and Singularity of Interval Matrices
cites work: Q4303969
cites work: Bounds on Eigenvalues of Interval Matrices
DOI: 10.1016/J.AMC.2010.09.066
description: scientific article; zbMATH DE number 5862460