Newton flows for real equations
type: Item
instance of: scholarly article
title: Newton flows for real equations
zbMATH Open document ID: 0665.58035
DOI: 10.1216/RMJ-1988-18-2-433
author: Heinz-Otto Peitgen
author: Klaus Schmitt
author: Michael Pruefer
published in: Rocky Mountain Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: The dynamical system \(N(x)=x-DG(x)^{-1}G(x),\) G: \(R^ n\to R^ n\) smooth, provided by the Newton method for \(G(x)=0\) is studied. For \(n=2\) and G rational the dynamics is understood in terms of the theory of Julia sets. Let S be the singular set of G and define the ``Julia-like'' set of N by \(J_ N=closure\{x\in R^ n| \quad N^ k(x)\in S,\quad some\quad k\}\) and the ``exploding'' set of N by \(E_ N=closure\{x\in R^ n| \quad \| N^ k(x)\| \to \infty,\quad as\quad k\to \infty \}.\) The authors give some evidence for the following conjecture (which is true for Newton's method for rational mappings of C): \(J_ N=E_ N\); and a particular model problem in \(R^ 2\) is discussed.
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 4088574
zbMATH Keywords: Newton flows
zbMATH Keywords: Julia sets
reviewed by: R. Schmid
MaRDI profile type: Publication
full work available at URL: rmj-1988-18-2-433
OpenAlex ID: W2019542349
description: scientific article; zbMATH DE number 4088574