Bounds for eigenvalues of matrix polynomials
type: Item
instance of: scholarly article
title: Bounds for eigenvalues of matrix polynomials
zbMATH Open document ID: 1055.15030
DOI: 10.1016/S0024-3795(01)00316-0
author: Nicholas J. Higham
author: Françoise Tisseur
published in: Linear Algebra and its Applications
publication date: 2003-02-05T00:00:00Z
review text: Let \(P(\lambda)= \lambda^m A_m+ \lambda^{m-1} A_{m-1}+\cdots+ A_0\) be a polynomial in \(\lambda\) with matrix coefficients. A great variety of upper and lower bounds are derived here for \(|\lambda|\), where \(\lambda\) is an eigenvalue of \(P\). The authors concentrate on bounds that are of practical use, and many of these generalize known inequalities. Some are based on 1, \(\infty\) and 2-norms of block companion matrices \(C_L\) and \(C_U\) whose blocks are constructed from the \(A_j\). Two of the bounds generalize earlier ones of Cauchy, Montel, Carmichael and Mason, and Kojima. Others involve the singular values of \(C_U\) and \(C_L\) and the characteristic polynomial of \(P\). The authors did numerical experiments to test many of their bounds. One involved a polynomial of degree 9 with \(5\times 5\) coefficient matrices having random entries from the normal \((0,1)\) distribution. The upper bounds were quite sharp, the lower bounds more variable. Two other experiments involved polynomials of degree 2 with much larger coefficient matrices, one arising from a differential equation and the other from a structural dynamics model. Some of the upper bounds for the dominant eigenvalue were quite good, but one was a severe overestimate. The authors remark that identifying bounds suited to particular classes of problems would be valuable.
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15A54
Mathematics Subject Classification ID: 65F15
zbMATH DE Number: 1864788
zbMATH Keywords: eigenvalues
zbMATH Keywords: matrix polynomials
zbMATH Keywords: upper and lower bounds
zbMATH Keywords: singular values
zbMATH Keywords: numerical experiments
describes a project that uses: JDQR
describes a project that uses: JDQZ
describes a project that uses: ARPACK
MaRDI profile type: Publication
Wikidata QID: Q56998807
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description: scientific article; zbMATH DE number 1864788