Polynomiography via Ishikawa and Mann Iterations

Abstract

The aim of this paper is to present some modifications of complex polynomial roots finding visualization process. In this paper Ishikawa or Mann iterations are used instead of the standard Picard iteration. Kalantari introduced the name polynomiography for that visualization process and the obtained images he called polynomiographs. Polynomiographs are interesting both from educational and artistic point of view. By the use of different iterations we obtain quite new polynomiographs that look aestheatically pleasing comparing to the ones from standard Picard iteration. As examples we present some polynomiographs for complex polynomial equation z^3 - 1 = 0, permutation and doubly stochastic matrices. We believe that the results of this paper can inspire those who may be interested in aesthetic patterns created automatically. They also can be used to increase functionality of the existing polynomiography software.