A characterisation of Newton maps

Arno Berger, University of Canterbury
Theodore P. Hill, Georgia Institute of Technology - Main Campus

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Copyright © 2006 Australian Mathematical Society.

NOTE: At the time of publication, the author Theodore Hill was not yet affiliated with Cal Poly.

Abstract

Conditions are given for a C^k 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, i.e. 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 T¹ there, and it is best possible as is demonstrated through examples.