Newton's method for the quadratic matrix equation
type: Item
instance of: scholarly article
title: Newton's method for the quadratic matrix equation
zbMATH Open document ID: 1117.65071
author: Yong-Hua Gao
published in: Applied Mathematics and Computation
publication date: 2007-01-11T00:00:00Z
review text: Quadratic matrix equations (QME) arise in many areas of scientific computing. In this paper, the author studies the quadratic matrix equation \({\mathcal L}(X)=AX^2+BX+C=0\), where \(A, B, C \in {\mathbb R}^{n\times n}\), and \((x^T B x)^2>4(x^T A x)(x^T C x),\,\forall x\not=0.\) These equations are connected with the overdamped quadratic eigenvalue problem in the analysis of damped structural systems and vibration problem. Some techniques for analysis and solution to the equation \({\mathcal L}(X)=0\), and other nonlinear problems have been addressed by many authors. In this paper, the author studies another different technique to solve the equation \({\mathcal L}(X)=0\), with the condition \((x^T B x)^2>4(x^T A x)(x^T C x),\,\forall x\not=0.\) It is readily seen that if the quadratic matrix equation \({\mathcal L}(X)=0\) has a nonsingular solution \(S\), \(S\) is also the solution of the nonlinear matrix equation (NME) \({\mathcal F}(X)=AX+CX^{-1}+B=0.\) The author applies Newton's method to the nonlinear matrix equation \({\mathcal F}(X)=0\) for computing the \textit{dominant solvent} and the \textit{minimal solvent} of the quadratic matrix equation \({\mathcal L}(X)=0\), because the \textit{minimal solvent} of the quadratic matrix equation \({\mathcal L}(X)=0\) is also the dominant solution of the nonlinear matrix equation \({\mathcal F}(X)=0\) (provided that the minimal solvent is nonsingular). The author derives a local convergence theorem and a semilocal convergence theorem for Newton's method. Also, numerical results are presented to show the feasibility and effectiveness of the method developed.
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 5115448
zbMATH Keywords: Quadratic matrix equation
zbMATH Keywords: nonlinear matrix equation
zbMATH Keywords: iteration method
zbMATH Keywords: convergence
zbMATH Keywords: quadratic eigenvalue problem
zbMATH Keywords: numerical results
reviewed by: Sonia Pérez Díaz
describes a project that uses: SQUINT
MaRDI profile type: Publication
full work available at URL: j.amc.2006.06.014
OpenAlex ID: W1969228025
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DOI: 10.1016/J.AMC.2006.06.014
description: scientific article; zbMATH DE number 5115448