Abstract and Applied Analysis

Volume 2015 (2015), Article ID 797594, 19 pages

http://dx.doi.org/10.1155/2015/797594

## Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods

Institute of Computer Science, University of Silesia, Będzińska 39, 41-200 Sosnowiec, Poland

Received 17 December 2014; Accepted 5 February 2015

Academic Editor: Naseer Shahzad

Copyright © 2015 Krzysztof Gdawiec et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A survey of some modifications based on the classic Newton’s and the higher order Newton-like root finding methods for complex polynomials is presented. Instead of the standard Picard’s iteration several different iteration processes, described in the literature, which we call nonstandard ones, are used. Kalantari’s visualizations of root finding process are interesting from at least three points of view: scientific, educational, and artistic. By combining different kinds of iterations, different convergence tests, and different colouring we obtain a great variety of polynomiographs. We also check experimentally that using complex parameters instead of real ones in multiparameter iterations do not destabilize the iteration process. Moreover, we obtain nice looking polynomiographs that are interesting from the artistic point of view. Real parts of the parameters alter symmetry, whereas imaginary ones cause asymmetric twisting of polynomiographs.

#### 1. Introduction

Polynomial root-finding has played a key role in the history of mathematics. It is one of the oldest and most deeply studied mathematical problems. In 2000 BC Babylonians solved quadratic equation (quadratics). Seventeen centuries later Euclid solved quadratics with geometrical construction. In 1539 Cardan gave complete solution to cubics. In 1699 Newton introduced numerical iteration for root-finding. About seventy years later Lagrange showed that polynomial of degree 5 or higher cannot be solved by the methods used for quadratics, cubics, and quartics. In 1799 Gauss proved the Fundamental Theorem of Algebra. 27 years later Abel proved the impossibility of generally solving equations of degree higher than 4. General root-finding method has to be iterative and can only be done approximately. Cayley in 1879 observed strange and unpredictable chaotic behaviour of the roots approximation process while applying Newton’s method to the equation in the complex plane. The solution of Caley’s problem was found in 1919 by Julia. Julia sets became an inspiration for the great discoveries in 1970s, the Mandelbrot set and fractals [1]. The last interesting contribution to the polynomials root finding history was made by Kalantari [2], who introduced the polynomiography. It defines the visualization process of the approximation of the roots of complex polynomials, using fractal and nonfractal images created via the mathematical convergence properties of iteration functions. An individual image is called a polynomiograph. Polynomiography combines both art and science aspects. As a method which generates nice looking graphics, it was patented by Kalantari in USA in 2005 [3].

It is known that any complex polynomial of degree having roots, according to the Fundamental Theorem of Algebra, can be uniquely defined by its coefficients : or by its zeros (roots) :

Iterative roots finding process can be obviously applied to both representations of . The polynomiographs are generated as the result of this process’ visualization. The degree of the polynomial defines the number of basins of attraction (root’s basin of attraction is an area of the complex plane in which each point is convergent to the root using the root finding method). Localizations of the basins can be controlled by changing the roots positions on the complex plane manually.

Usually, polynomiographs are coloured 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, 4].

Fractals and polynomiographs are generated by iterations. Fractals are self-similar, have complicated and nonsmooth structure, and are not dependent on a resolution. Polynomiographs are different. Their shape can be controlled and designed in a more predictable way in opposition to fractals. Generally, fractals and polynomiographs belong to different classes of graphical objects.

Summing up, polynomiography can be treated as a visualization tool based on the root finding process. It has many possible applications in education, math, sciences, art, and design [2].

In [5] the authors used Mann and Ishikawa iterations instead of the standard Picard iteration to obtain some generalization of Kalantari’s polynomiography and presented some polynomiographs for the cubic equation , permutation, and double stochastic matrices. Latif et al. in [6], using the ideas from [5], have used the -iteration in polynomiography. Earlier, the other types of iterations have been used in [7] for superfractals and in [8] for fractals generated by IFS. Julia sets and Mandelbrot sets [9] and the antifractals [10] have been also investigated using Noor iteration instead of the standard Picard iteration.

The paper is organised as follows. In Section 2 different kinds of iterations are presented. Section 3 presents the known root finding methods, starting from the known Newton’s method up to the different generalizations of it. Section 4 treats different convergence tests used in iteration processes together with their modifications. In Section 5 the colouring methods of polynomiographs are introduced. Section 6 summarizes the theory of polynomiograph generation. As a result the full algorithm of polynomiograph generation is given. Section 7 presents many polynomiographs obtained experimentally as the result of the proposed algorithm. In Section 8 the time complexity of this algorithm is discussed. Section 9 concludes the paper and shows the future directions.

#### 2. Iterations

Obviously, the equation of the form can be equivalently transformed into a fixed point problem , where is some operator [11]. Then, by applying the approximate fixed point theorem one can get information on existence, or sometimes both on existence and uniqueness, of the fixed point that is the solution of this equation.

Let be a complete metric space and a self-map on . The set is the set of all fixed points of . Many iterative processes have been described for the approximation of fixed points in the ample literature [12–19]. We recall below some iteration processes known from the literature. Assume that each iteration process starts from any initial point .(i)The standard Picard iteration [20] introduced in 1890 is defined as (ii)The Mann iteration [16] was defined in 1953 as where for all .(iii)The Ishikawa iteration [13] was defined in 1974 as a two-step process: where and for all .(iv)The Noor iteration [17] was defined in 2000 as a three-step process: where and for all .(v)The Suantai iteration [19] was defined in 2005 as a three-step iteration process with five parameters: where , , and for all and .(vi)In 2007 Agarwal et al. in [21] introduced the -iteration: where and for all .(vii)The SP iteration [18] was defined in 2011 as the following three-step process: where and for all .(viii)In 2012 Chugh et al. introduced the CR iteration in [22]: where for all and .(ix)In 2013 Khan iteration [15] was defined as the following process: where for all .(x)In 2013 Karakaya et al. defined very general three-step iteration process with five parameters in [14]: where , , and for all and .(xi)In 2014 Gürsoy and Karakaya introduced the Picard-S iteration in [23]: where and for all .

The standard Picard iteration is used in the Banach Fixed Point Theorem [12] to obtain existence of the fixed point of the operator . Fixed point approximation is found under additional assumptions on the space that it has to be a Banach one and the mapping has to be contractive. The Mann [16], Ishikawa [13], and other iterations [12, 14, 15, 17–19] allow the weakening of the assumptions on the mapping and generally allow the approximation of fixed points. The dependencies among the presented types of iterations are shown in Figure 1.