Estimating the radius of an attraction ball

7 years ago

(original), Fixed point theory, inexact/perturbed iterations, ISI/JCR, local convergence, Newton method, nonlinear systems in Rn, Numerical Analysis, paper, solving F(x)=0 iteratively, successive approximations

Abstract

Given a nonlinear mapping $G:D\subseteq \mathbb{R}^n\rightarrow \mathbb{R}^n$ differentiable at a fixed point $x^\ast$ , the Ostrowski theorem offers the sharp sufficient condition $\rho(G′(x^\ast))<1$ for $x^\ast$ to be an attraction point, where $\rho$ denotes the spectral radius. However, no estimate for the size of an attraction ball is known.

We show in this note that such an estimate may be readily obtained in terms of $\|G^\prime(x^\ast)\|<1$ (with $\|\cdot \|$ an arbitrary given norm) and of the Hölder (in particular Lipschitz) continuity constant of $G’$ .

An elementary example shows that this estimate may be sharp. Our assumptions do not necessarily require $G$ to be of contractive type on the whole estimated ball.

Authors

Emil Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Keywords

fixed point; attraction points; attraction ball; local convergence; convergence order.

Cite this paper as:

E. Cătinaş, Estimating the radius of an attraction ball, Appl. Math. Lett., 22 (2009) no. 5, pp. 712-714.

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preprint version of the paper.

About this paper

Journal

Applied Mathematics Letters

Publisher Name

Elsevier

DOI

10.1016/j.aml.2008.08.007

Print ISSN

0893-9659

Online ISSN

Google Scholar citations

2009