Fractal Characteristics of Newton's Method on Polynomials

Abstract

In this report, we present a simple geometric generation principle for the fractal that is obtained when applying Newton's method to find the roots of a general complex polynomial with real coefficients. For the case of symmetric polynomials , the generation mechanism is derived from first principles. We discuss the case of a general cubic and are able to give a description of the arising fractal structure depending on the coefficients of the cubic. Special cases are analysed and their characteristics, including scale factors and an approximate fractal dimension, are derived. The theoretical results are confirmed via computational experiments. An application of the theory in turbulence modelling is presented.