How to decrease the combinatory complexity

TL;DR: In this paper, the authors consider the problem of approximating a nonlinear scalar equation with one-point iteration without memory and define the sequence of successive approximations for a given integer s, s.

Abstract: Introduction We consider the solution of a nonlinear scalar equation (1) f(x) = 0, where f: Dc£-"-C . Assume that f is analytic in a neighbourhood of a simple zero a , f(a) = 0 4 f' (oc). We approximate ot by an iteration. Suppose that we can compute the standard information on f, i.e. (2) 31 (f,x) = {f(x)t f' (x),..., f(s)(x)} for a given integer s, s We deal with a one-point iteration without memory <p which defines the sequence of successive approximations {xj_} by