Global aspects of the continuous and discrete Newton method: A case study
type: Item
instance of: scholarly article
title: Global aspects of the continuous and discrete Newton method: A case study
zbMATH Open document ID: 0669.65038
DOI: 10.1007/BF00047504
author: Heinz-Otto Peitgen
author: Klaus Schmitt
author: Michael Pruefer
published in: Acta Applicandae Mathematicae
publication date: 1988-01-01T00:00:00Z
review text: In this paper and the related paper of \textit{D. Saupe} [ibid. 13, No.1/2, 59-80 (1988; reviewed above)] the qualitative convergence properties of Newton's method, some damped versions of Newton's method, and the continuous Newton's method are studied via the Julia set theory. In particular, the various forms of Newton's method are applied to complex polynomials, boundary value problems, and their corresponding discretizations. The present paper being to some extend experimental in nature consists of two lengthy chapters. Chapter 2 is concerned with the study of the Newton flow \((I): \dot x=[DG(x(t))]^{-1}G(x(t)),\) \(x(0)=x_ 0\) for general maps and for maps including a bifurcation parameter. Chapter 3 deals with the qualitative performance of Newton's method when G is a standard discretization of a boundary value problem. It deals with questions concerning the phase portraits, the role of the singular set, and Julia-like sets. Chapter 3 contains many insightful computer graphic diagrams.
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 4096692
zbMATH Keywords: damped Newton method
zbMATH Keywords: fractal basin boundary
zbMATH Keywords: convergence
zbMATH Keywords: continuous Newton's method
zbMATH Keywords: Julia set theory
zbMATH Keywords: complex polynomials
zbMATH Keywords: Newton flow
zbMATH Keywords: bifurcation
zbMATH Keywords: phase portraits
zbMATH Keywords: singular set
zbMATH Keywords: computer graphic diagrams
reviewed by: Eugene L. Allgower
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00047504
OpenAlex ID: W4239862943
description: scientific article; zbMATH DE number 4096692