Towards a Global Convergence Theory for Newton's Method

In this paper, we present a global convergence theory for Newton's Method in n dimensions for functions of polynomial character (generic or truncated Taylor series), augmenting the classical Newton-Kantorovich results on quadratic convergence. It is shown that there exist ve distinct, quantiiable convergence states, which can be described by a nite state machine. From this convergence description, it is also possible to deduce a strictly monotonically decreasing measure that constitutes a `natural metric' of Newton's Method. Implications and possible generalisations are stated, as well as a variety of examples.