Kyle Kneisl

Julia sets for the super-Newton method, Cauchy's method, and Halley's method.

We study numerically and dynamically three cubically convergent iterative root-finding algorithms, namely Cauchy's method, the super-Newton method, and Halley's method. Using the concept of a universal Julia set (motivated by the results of McMullen), we establish that these algorithms converge when applied to any quadratic with distinct roots. We give examples showing the existence of attracting periodic orbits not associated to a root for the super-Newton method and Halley's method applied to cubic polynomials. We include computer plots showing the dynamic structure for each algorithm applied to a variety of polynomials. (c) 2001 American Institute of Physics.