Published in Australian & New Zealand Industrial and Applied Mathematics Journal, Volume 48, January 1, 2006, pages 211-223. Copyright © 2006 Australian Mathematical Society.
NOTE: At the time of publication, the author Theodore Hill was not yet affiliated with Cal Poly.
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, 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.