Numerical validation for an inverse matrix eigenvalue problem
type: Item
instance of: scholarly article
title: Numerical validation for an inverse matrix eigenvalue problem
zbMATH Open document ID: 0813.65076
DOI: 10.1007/BF02307382
author: A. Gienger
author: Günter Mayer
author: Gotz Alefeld
published in: Computing
publication date: 1994-12-20T00:00:00Z
review text: The authors consider the following problem (called the additive inverse eigenvalue problem): Given \(n + 1\) real symmetric \(n \times n\) matrices \(A_ i\), \(i = 0, 1, \dots, n\), and given \(n\) real numbers \(\lambda_ 1 < \lambda_ 2 < \dots < \lambda_ n\), prove that there are \(n\) real numbers \(c^*_ i\), \(i = 1, \dots, n\), such that the matrix \(A(c) : = A_ 0 + \sum^ n_{i=1} c_ i A_ i\), \(c : = (c_ i) \in \mathbb{R}^ n\), has for \(c_ i = c^*_ i\), \(i = 1,\dots,n\), the prescribed numbers \(\lambda_ i\) as eigenvalues. The authors solve this problem by means of interval computation. The proposed algorithm is based on Newton's method using a new criterion for terminating the iteration, in order to get an approximation \(\widetilde c\) of \(c^* = (c^*_ i)\). The existence of \(c^*\) using one step of the Krawczyk method is verified. Two numerical examples are reported.
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 65G30
zbMATH DE Number: 704629
zbMATH Keywords: enclosure
zbMATH Keywords: prescribed eigenvalue
zbMATH Keywords: additive inverse eigenvalue problem
zbMATH Keywords: interval computation
zbMATH Keywords: Newton's method
zbMATH Keywords: Krawczyk method
zbMATH Keywords: numerical examples
reviewed by: G. Dimitriu
describes a project that uses: PASCAL-XSC
MaRDI profile type: Publication
cites work: Berechenbare Fehlerschranken für ein Eigenpaar unter Einschluß von Rundungsfehlern bei Verwendung des genauen Skalarprodukts
cites work: Efficient Numerical Validation of Solutions of Nonlinear Systems
cites work: Q3345690
cites work: Q4851640
cites work: Numerical solution of the inverse algebraic eigenvalue problem
cites work: Inverse eigenvalue problems
cites work: Q3783418
cites work: The Formulation and Analysis of Numerical Methods for Inverse Eigenvalue Problems
cites work: A Unified Approach to Enclosure Methods for Eigenpairs
cites work: Q4321539
cites work: Q4860286
cites work: Q3996567
cites work: Q3857636
full work available at URL: https://doi.org/10.1007/bf02307382
OpenAlex ID: W179652530
description: scientific article; zbMATH DE number 704629