Polynomiography via an iterative method corresponding to Simpsons 13 rule

Abstract

The aim of this paper is to present some artwork produced via polynomiography of a few complex polynomials and a few special polynomials arising in science as well as a few considered to arrive at beautiful but anticipated designs. In this paper an iterative method corresponding to Simpson's 1 3 rule is used instead of Newton's method. The word "polynomiography" coined by Kalantari for that visualization process. The images obtained are called polynomiographs. Polynomiographs have importance for both the art and science aspects. By using an iterative method corresponding to Simpson's 1 3 rule, we obtain quite new nicely looking polynomiographs that are different from Newton's method. Presented examples show that we obtain very interesting patterns for complex polynomial equations, permutation matrices, doubly stochastic matrices, Chebyshev polynomial, polynomial arising in physics and Alexander polynomial in knot theory. We believe that the results of this paper enrich the functionality of the existing polynomiography software. c 2016 All rights reserved.

Study of Dynamical Behavior and Stability of Iterative Methods for Nonlinear Equation with Applications in Engineering

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