A modification of Newton's method for analytic mappings having multiple zeros
type: Item
instance of: scholarly article
title: A modification of Newton's method for analytic mappings having multiple zeros
zbMATH Open document ID: 0933.65049
published in: Computing
publication date: 2000-03-20T00:00:00Z
review text: A modification of Newton's method for iteratively computing singular zero points of analytic functions is studied. Under mild assumptions the method converges quadratically, without a priori information concerning the nature of the singularity. The method generalizes to systems \textit{H. Van de Vel}'s iterative method for finding zero points of an analytic map in a single variable [Computing 14, 167-171 (1975; Zbl 0301.65028)]. The latter method determines the zero point \(z^*\) superlinearly and also the multiplicity \(\mu_{z^*}(f)\) of \(z^*\). The present generalization gives approximations of zero points \(z^*\in\mathbb{C}^n\) and also of the orders \(k_1,\cdots,k_n\) of the homogeneous principal parts of \(f\) at \(z^*\). That is, \(f_j(z)=\sum_{|\alpha|\geq k_j}c_{j,\alpha}(z-z^*)^{\alpha}\) where \(\alpha\) is a multi-index. Several numerical examples are given which illustrate the effectiveness of the approach.
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 32A60
Mathematics Subject Classification ID: 65E05
zbMATH DE Number: 1309795
zbMATH Keywords: Newton's method
zbMATH Keywords: multiple zeros
zbMATH Keywords: singular zero points
zbMATH Keywords: analytic functions
zbMATH Keywords: systems
zbMATH Keywords: numerical examples
author: Peter Kravanja
author: Ann Haegemans
reviewed by: Eugene L. Allgower
MaRDI profile type: Publication
full work available at URL: s006070050017
OpenAlex ID: W2099505907
DOI: 10.1007/S006070050017
description: scientific article; zbMATH DE number 1309795