NEWTON'S VERSUS HALLEY'S METHOD: A DYNAMICAL SYSTEMS APPROACH

We compare the iterative root-finding methods of Newton and Halley applied to cubic polynomials in the complex plane. Of specific interest are those "bad" polynomials for which a given numerical method contains an attracting cycle distinct from the roots. This implies the existence of an open set of initial guesses whose iterates do not converge to one of the roots (i.e. the numerical method fails). Searching for a set of bad parameter values leads to Mandelbrot-like sets and interesting figures in the parameter plane. We provide some analytic and geometric arguments to explain the contrasting parameter plane pictures. In particular, we show that there exists a sequence of parameter values λn for which the corresponding numerical method has a superattracting n cycle. The λn lie at the centers of a converging sequence of Mandelbrot-like sets.