Fixed points, zeros and Newton's method
type: Item
instance of: scholarly article
title: Fixed points, zeros and Newton's method
zbMATH Open document ID: 1095.65047
DOI: 10.1007/3-540-37660-7
author: Jean-Pierre Dedieu
published in: Mathématiques \& Applications (Berlin)
publication date: 2006-03-13T00:00:00Z
review text: In the introductory Chapter 1 of the book under review, the author states that the book has its origin in a university course about methods of solving systems of nonlinear equations; but, while making up the redaction of that course more topics have been added, and so the final result is a book containing much more than the things that he had originally in mind. Hence, the author has produced a book of 196 pages, structured as follows: Chapter 1: Introduction (pp. 1--3); Chapter 2: Fixed points (pp. 5--74); Chapter 3: Newton's method (pp. 75--110); Chapter 4: Newton's method for underdetermined systems (pp. 111--144); Chapter 5: The method of Newton-Gauss for overdetermined systems (pp. 145--175); Chapter 6: Appendices (pp. 177--190). The book ends with a list of 57 references (pp. 191--193) and an index (pp. 195--196). Returning to the introductory Chapter 1, the author states that the book is intended for students doing higher level studies in mathematics, and for researchers. What is needed for reading it, he says, is a good knowledge of linear algebra, general topology and differential calculus, and some knowledge of functional analysis and of complex variables. The potential reader should also know that the two general frames in which the results are stated are either Banach spaces or Hilbert spaces (a few results are stated in complete metric spaces). In order to make the book in some sense ``self-contained'', the author has added the appendices in Chapter 6, containing a.o. some definitions and results about differential calculus on Banach spaces, on Hilbert spaces and on Euclidean spaces. The author describes the contents of the book as consisting of three parts. The first part is devoted to fixed points (chapter 2). By way of example, for differentiable maps on Banach spaces three categories of fixed points are considered: attractive, repulsive and hyperbolic fixed-points (connected to the theorem of Grobman-Hartman when the map is nonlinear); conntected to…
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 65-01
Mathematics Subject Classification ID: 37C25
Mathematics Subject Classification ID: 49M25
Mathematics Subject Classification ID: 58C15
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 5011754
zbMATH Keywords: systems of nonlinear equations
zbMATH Keywords: textbook
zbMATH Keywords: fixed points
zbMATH Keywords: Newton's method
zbMATH Keywords: underdetermined systems
zbMATH Keywords: method of Newton-Gauss
zbMATH Keywords: Banach spaces
zbMATH Keywords: Hilbert spaces
zbMATH Keywords: Euclidean spaces
zbMATH Keywords: generalised inverse
zbMATH Keywords: alpha-theory
reviewed by: Gilbert Crombez
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/3-540-37660-7
OpenAlex ID: W614309271
description: scientific article; zbMATH DE number 5011754