Newton's method's basins of attraction revisited

In this paper, we revisit the chaotic number of iterations needed by Newton's method to converge to a root. Here, we consider a simple modified Newton method depending on a parameter. It is demonstrated using polynomiography that even in the simple algorithm the presence and the position of the convergent regions, i.e. regions where the method converges nicely to a root, can be complicatedly a function of the parameter. The study of root-finding to an equation through an iterative function has a very long history [1]. An algorithm, which is a cornerstone to the modern study of root-finding algorithms was made by Newton through his 'method of fluxions'. Later on this method was polished by Rapshon to produce what we now know as the Newton–Raphson method [1–3]. Suppose we want to solve a nonlinear equation f ðzÞ ¼ 0 numerically, where z 2 R and the function f : R ! R is at least once differentiable. Starting from some z 0 , the Newton–Raphson method uses the iterations: