Polynomiography for Square Systems of Equations with Mann and Ishikawa Iterations

Abstract

In this paper we propose to replace the standard Picard iteration in the Newton--Raphson method by Mann and Ishikawa iterations. This iteration's replacement influence the solution finding process that can be visualized as polynomiographs for the square systems of equations. Polynomiographs presented in the paper, in some sense, are generalization of Kalantari's polynomiography from a single polynomial equation to the square systems of equations. They are coloured based on two colouring methods: basins of attractions with different colours for every real root and colouring dependent on the number of iterations. Possible application of the presented method can be addressed to computer graphics where aesthetic patterns can be used in e.g. texture generation, animations, tapestry design.