Visualization of polynomiography from new higher order iterative methods

P olynomials are one of the most significant objects in many fields of mathematics. Polynomial root-finding has played a key role in the history of mathematics [2]. It is one of the oldest and most deeply studied mathematical problems. The last interesting contribution to the polynomials root finding history was made by Kalantari [2], who introduced the polynomiography as a method which generates nice looking graphics [3,4]. Polynomiography is defined to be the art and science of visualization in approximation of the zeros of complex polynomials, via fractal and non fractal images created using the mathematical convergence properties of iteration functions [5]. An individual image is called a " polynomiograph ”and polynomiography combines both art and science aspects [6,7]. Polynomiography gives a new way to solve the ancient problem by using new algorithms and computer technology [8]. Polynomiography is based on the use of one or an infinite number of iteration methods formulated for the purpose of approximation of the root of polynomials e.g. Newton’s method, Halley’s method etc. The “ polynomiographer ” can control the shape and designed in a more predictable way by using different iteration methods to the infinite variety of complex polynomials [9]. According to Fundamental Theorem of Algebra, any complex polynomial with complex coefficients {an, an−1, ..., a1, a0}: p(z) = anz + an−1z + ... + a1z + a0 (1)


Introduction
P olynomials are one of the most significant objects in many fields of mathematics. Polynomial root-finding has played a key role in the history of mathematics [2]. It is one of the oldest and most deeply studied mathematical problems. The last interesting contribution to the polynomials root finding history was made by Kalantari [2], who introduced the polynomiography as a method which generates nice looking graphics [3,4]. Polynomiography is defined to be the art and science of visualization in approximation of the zeros of complex polynomials, via fractal and non fractal images created using the mathematical convergence properties of iteration functions [5]. An individual image is called a " polynomiograph "and polynomiography combines both art and science aspects [6,7].
Polynomiography gives a new way to solve the ancient problem by using new algorithms and computer technology [8]. Polynomiography is based on the use of one or an infinite number of iteration methods formulated for the purpose of approximation of the root of polynomials e.g. Newton's method, Halley's method etc. The " polynomiographer " can control the shape and designed in a more predictable way by using different iteration methods to the infinite variety of complex polynomials [9].
According to Fundamental Theorem of Algebra, any complex polynomial with complex coefficients {a n , a n−1 , ..., a 1 , a 0 }: p(z) = a n z n + a n−1 z n−1 + ... + a 1 z + a 0 (1) or by its zeros (roots) {r 1 , r 2 , ..., r n−1 , r n } : of degree n has n roots (zeros) which may or may not be distinct. The degree of polynomial describes the number of basins of attraction and placing roots on the complex plane manually localization of basins can be controlled. Usually, polynomiographs are colored based on the number of iterations needed to obtain the approximation of some polynomial root with a given accuracy and a chosen iteration method. The description of polynomiography, its theoretical background and artistic applications are described in [2][3][4]. In this paper, we aim to present polynomiography via iterative methods presented in [1].

Algorithm 2.
Let p(z) be the complex polynomial, then which is algorithm (2) for solving nonlinear complex equations.

Algorithm 3.
Let p(z) be the complex polynomial, then which is algorithm (3) for solving nonlinear complex equations.
In above algorithms, z o ∈ C is a starting point. The sequence {z n } ∞ n=0 is called the orbit of the point z o converges to a root z * of p then, we say that z o is attracted to z * . A set of all such starting points for which {z n } ∞ n=0 converges to root z * is called the basin of attraction of z * .

Convergence test
In the numerical algorithms that are based on iterative processes we need a stop criterion for the process, that is, a test that tells us that the process has converged or it is very near to the solution [6][7][8][9]. This type of test is called a convergence test. Usually, in the iterative process that use a feedback, like the root finding methods, the standard convergence test has the following form: where z n+1 and z n are two successive points in the iteration process and ε > 0 is a given accuracy. In this paper we also use the stop criterion (2).

Polynomiographs
In this section we present some examples of polynomiographs for different complex polynomials equation p(z) = 0 and some special polynomials using algorithms proposed in [1]. The different colors of a images depend upon number of iterations to reach a root with given accuracy ε = 0.001. One can obtain infinitely many nice looking polynomiographs by changing parameter k, where k is the upper bound of the number of iterations. Example 1. The polynomiographs for z 2 − 1 = 0 by using Algorithms 1, 2 and 3 are presented here in Figures  1, 2 and 3 respectively.                  Author Contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Conflicts of Interest: "The authors declare no conflict of interest."