Computing isolated singular solutions of polynomial systems: case of breadth one
type: Item
instance of: scholarly article
zbMATH Open document ID: 1247.65065
DOI: 10.1137/110827247
published in: SIAM Journal on Numerical Analysis
publication date: 2012-05-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1008.0061
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 65H04
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 13P05
zbMATH DE Number: 6030218
zbMATH Keywords: root refinement
zbMATH Keywords: isolated singular solution
zbMATH Keywords: regularized Newton iteration
zbMATH Keywords: local dual space
zbMATH Keywords: quadratic convergence
zbMATH Keywords: polynomial system
describes a project that uses: Maple
MaRDI profile type: Publication
OpenAlex ID: W2091853326
arXiv ID: 1008.0061
title: Computing isolated singular solutions of polynomial systems: case of breadth one
author: Nan Li
author: Lihong Zhi
review text: Let us consider an approximate singular solution \(\hat{\eta}\) of a polynomial system \(F=\{f_1, \dots, f_n\}\) satisfying \(\|\hat \eta- \hat \eta_{{e}}\|=\varepsilon\), where \(\hat{\eta}_{{e}}\) denotes the isolated exact singular solution of \(F\), the positive number \(\varepsilon\) is small enough such that there are no other solutions of \(F\) nearby, and assume that the corank of the Jacobian matrix \(F'(\hat{\eta}_{{e}})\) is one. In view to restore the quadratic convergence of the Newton method, the authors apply one regularized Newton iteration to obtain a new approximation \(\hat \eta+\hat \theta\), and then compute the approximate null vector \(\vartheta_1\) of the Jacobian \(F'(\hat \eta+\hat \theta)\) which gives a generalized Newton direction, and the step length \(\delta\) is obtained by solving a linear system. They show that NEWLINE\[NEWLINE\|\hat \eta+\hat \theta+ \delta \vartheta_1- \hat \eta_{{e}}\| = O(\varepsilon^2).NEWLINE\]NEWLINE The size of matrices involved in the algorithm is bounded by \(n\times n\), and the method has been implemented in Maple.
reviewed by: L. P. Castro
description: scientific article; zbMATH DE number 6030218