Newton method under weak Lipschitz continuous derivative in Banach spaces.
type: Item
instance of: scholarly article
title: Newton method under weak Lipschitz continuous derivative in Banach spaces.
zbMATH Open document ID: 1042.65044
DOI: 10.1016/S0096-3003(02)00215-1
author: Zhengda Huang
published in: Applied Mathematics and Computation
publication date: 2003-09-09T00:00:00Z
review text: Let \(f: X \to Y\) be a Fréchet differentiable nonlinear operator from a Banach space \(X\) into a Banach space \(Y\). The Newton method for solving the equation \(f(x) = 0\) is described by the iterative formula \(x^{n+1} = x^n - (f'(x^n))^{-1} f(x^n),\) \(n>0\), provided that \((f'(x^n))^{-1}\) exists in each step. \textit{A. Ostrovski} [Comment. Math. Helv. 9, 79--103 (1937; Zbl 0015.36403)] and \textit{L. Kantorovich} [On Newton's method, Trudy. Math. Inst. Steklov 28, 104--144 (1949)] used majorant function techniques to determine whether the iterative sequence defined by this method, starting from some pivot value \(x^0\) converges and at the next step get a theoretical error estimates which reflects the convergent order two of the sequence of iterations. In this paper the results of Ostrovski and Kantorovich are generalized to weaker conditions for \(f\) and \(f''\). A relaxed Kantorovich-type condition to guarantee the convergence is obtained. It is also demonstrated, that the conditions of Kantorovich and {those of \textit{S. Smale} [Proc. Symp. Honor G. S. Young, Laramie/Wyo. 1985, 185--196 (1986; Zbl 0613.65058)] are special cases of the developed ones. The relations between the new condition and the old ones are described.}
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 1978068
zbMATH Keywords: Newton method
zbMATH Keywords: generalized Kantorovich condition
zbMATH Keywords: Smale condition
zbMATH Keywords: weak Lipchitz condition
zbMATH Keywords: Banach space
zbMATH Keywords: majorant function techniques
zbMATH Keywords: convergence
reviewed by: Jiří Vaníček
MaRDI profile type: Publication
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full work available at URL: s0096-3003(02)00215-1
OpenAlex ID: W1987694004
description: scientific article; zbMATH DE number 1978068