Residual methods for the large-scale matrix \(p\)th root and some related problems
type: Item
instance of: scholarly article
title: Residual methods for the large-scale matrix \(p\)th root and some related problems
zbMATH Open document ID: 1200.65032
author: Braulio De Abreu
author: M. Raydan
published in: Applied Mathematics and Computation
publication date: 2010-10-25T00:00:00Z
review text: The authors present and analyze a low-cost residual algorithm for computing the matrix \(p\)th root, \(X\), of a nonlinear matrix equation \(X^p-A=0\), where \(A\in{\mathbb R} ^{n\times n}\) is symmetric and positive definite and \(p\geq 2\). This is the best possible scenario for Smith's method [\textit{M. I. Smith}, SIAM J. Matrix Anal. Appl. 24, No.~4, 971--989 (2003; Zbl 1040.65038)]. This algorithm has local convergence properties in its pure form and global convergence in its globalized form. It is more efficient and requires fewer iterations for convergence when \(p\) is small since the condition number of the derivative of the residual map, \(X^p-A\), increases with \(p\). It is also robust in general and suitable for large-scale problems for which high accuracy is not required.
reviewed by: Rémi Vaillancourt
Mathematics Subject Classification ID: 65F30
zbMATH DE Number: 5804921
zbMATH Keywords: nonlinear matrix equations
zbMATH Keywords: \(p\)th roots of matrices
zbMATH Keywords: residual methods
describes a project that uses: mftoolbox
MaRDI profile type: Publication
full work available at URL: j.amc.2010.06.002
OpenAlex ID: W1990954078
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DOI: 10.1016/J.AMC.2010.06.002
description: scientific article; zbMATH DE number 5804921