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A characterisation of Newton maps

Published online by Cambridge University Press:  17 February 2009

A. Berger  and
T. P. Hill
A. Berger
Affiliation:
Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand; e-mail: amo.berger@canterbury.ac.nz.
T. P. Hill
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, USA; e-mail: hill@math.gatech.edu.
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Abstract

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Conditions are given for a Ck map T to be a Newton map, that is, the map associated with a differentiable real-valued function via Newton's method. For finitely differentiable maps and functions, these conditions are only necessary, but in the smooth case, that is, for k = ∞, they are also sufficient. The characterisation rests upon the structure of the fixed point set of T and the value of the derivative T′ there, and it is best possible as is demonstrated through examples.

Keywords

Newton's methodNewton mapfixed point setattracting fixed point
Type
Research Article
Information
The ANZIAM Journal , Volume 48 , Issue 2 , October 2006 , pp. 211 - 223
Copyright
Copyright © Australian Mathematical Society 2006

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