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. Thin implies the existence of an open set of initial guesses whose iterates do not converge 1 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 chat there exists a sequence of parameter values lambda(n) for which the corresponding numerical method has a superattracting n cycle. The lambda(n) lie at the centers of a converging sequence of Mandelbrot-like sets.