Polynomiography via the Hybrids of Gradient Descent and Newton Methods with Mann and Ishikawa Iterations

Abstract

In this paper two hybrids of two algorithms, the Newton method and the Gradient Descent method, are presented in order to create polynomiographs. The first idea combines two methods as a convex combination, the second one as the two-step process. A polynomiograph is an image obtained as the result of roots finding method of a polynomial on a complex plain. In this paper the polynomiographs are also modified by using Mann and Ishikawa iterations instead of the standard Picard iteration. Coloring images by iterations reveals dynamic behavior of the used root-finding process and its speed of convergence. The paper joins, combines and modifies earlier results obtained by Kalantari, Zhang et al. and the authors. We believe that the results of the paper can inspire those who are interested in aesthetic patterns created automatically. They can also be used to increase functionality of the existing polynomiography software.