Dynamics of RK Iteration and Basic Family of Iterations for Polynomiography

Abstract

In this paper, we propose some modifications of the basic family of iterations with a new four-step iteration called RK iteration and its s-convexity. We present some graphical examples showing the dynamics of the new iteration in the colouring and shapes of the obtained polynomiographs compared to the ones from the basic family only. Moreover, the computational results reveal that the value of s in the s-convex combination of the RK iteration has a significant impact on the time taken by the iteration process for approximating the roots of the polynomials. The obtained results are interesting from an artistic and computational point of view.

1. Introduction

Polynomials are considered one of the most significant objects in many fields of mathematics. The problem of finding the roots of polynomials has been revived and expanded by Kalantari after his breakthrough research on polynomiography, which has established an important relationship between mathematics and arts. Polynomiography is defined as the art and science of visualisation in the approximation of the roots of polynomials using iteration functions (see [1,2]). Graphics obtained via polynomiography are called polynomiographs. These are distinctly different from fractals (a term coined by the well-known Mandelbrot [3]), which also refer to images from iterations in computer science. Fractals are very difficult to control efficiently and their shapes are completely defined by the input data, e.g., by the coefficient of an iterated function. On the other hand, polynomiographs are quite different. Their shapes can be controlled and designed in a more predictable way, in opposition to the typical fractals.

While polynomials remain to be fundamental entities in mathematics and other sciences, polynomiography has shown a systemic development of algorithms revealing the considerable visual beauty behind solving polynomial equations. It uses sophisticated mathematical algorithms on a computer to create beautiful polynomiographs; with proper software development, it turns the polynomial root-finding problem upside down and into a medium of expression, art, design, science, education, innovation, discovery, creativity, and many more (see, e.g., [2,4]).

According to the Fundamental Theorem of Algebra, a complex polynomial of degree n with complex coefficient $\{a_n,a_{n-1},\cdots ,a_0\}$ defined by

$$p\left(z\right)=a_n{z}^n+a_{n-1}{z}^{n-1}+\cdots +a_{1}z+a_0,$$

has n roots, which may be distinct or repeated. Such a polynomial can also be defined by its zeros (or roots) $\{r_1,r_2,\cdots ,r_n\},$ i.e.,

$$p\left(z\right)=(z-r_1)(z-r_2)\cdots (z-r_n).$$

The degree of the polynomial describes the number of basins of attraction that it has. Localisation of the basins can be controlled by placing roots on the complex plane manually. The chosen roots define the polynomial for which some iteration process has to be performed to find its zeros. Usually, polynomiographs are coloured based on the number of iterations needed to obtain an approximation of some polynomial root with a given degree of accuracy and the iteration method chosen. A detailed description of polynomiography, its theoretical background, and it artistic applications can be found in [2,4].

In recent years, researchers have studied several methods for finding the roots of polynomials [5]. One of the most common methods for finding the roots of polynomials is the so-called Newton’s method, which dates back to over three hundred years ago. Newton’s method solves nonlinear equations of the form $p\left(x\right)=0$ using the following iterative process: Given $x_0,$ calculate $x_{n+1}$ via the formula

$$x_{n+1}=x_n-\frac{p\left(x_n\right)}{p^{\prime }\left(x_n\right)},$$

(1)

where $p^{\prime }\left(x_n\right)$ is the derivative of p at $x_n.$ While Newton’s method shows accurate approximation for polynomials with degree 2, its convergence can be very complicated for polynomials with degree 3 or above [2,6]. Newton’s method has been generalised in many ways by different authors; see, for example, ref. [7]. Moreover, some whole families of iterations, such as the basic family of iterations [2], Euler–Schöder [2], Lofti et al. [8,9], Cordero et al. [10], and Jolaoso and Khan [11], have been proposed.

In [12], Gdawiec et al. presented a survey of some modifications of Newton’s method with non-standard iterations for generating polynomiographs. They used different iteration processes from the fixed point theory, such as Mann, Ishikawa, Noor, Khan, S, $SP$, $CR$, Suantai, and Karakaya iteration methods. Later, in [13], Kang et al. used the S-iteration instead of the Picard (standard) iteration, which coincides with Newton’s method. Moreover, Gdawiec and Kotarski [14] extended the list of iterations to several iteration processes from the fixed point theory. Rafiq et al. [15] proposed some implicit schemes such as Jungck–Mann and Jungck–Ishikawa with Newton’s method for generating polynomiographs. Recently, Gdawiec et al. [16] considered the basic family of iterations with S-iteration and s-convexity for polynomiography.

In this paper, we propose a new modification of the basic family of iterations using a four-step RK iteration and its s-convexity for polynomiography. The RK iteration was recently introduced by Ritika and Khan [17] for approximating fixed points of certain nonlinear mappings in metric spaces. Some of the peculiarities of the RK iteration are that it requires a smaller number of iterations to converge to the fixed point of the underlying mapping and thus converges faster than many known iterations, such as the Mann, Ishikawa, Noor, S, $SP$, $CR$, Suantai, Güsoy, and Karakaya iterations (see [17]). Using this new process, we obtained several polynomiographs with interesting dynamical properties for different examples of complex polynomials. Moreover, we give some remarks explaining the dynamics of the obtained graphics. These experiments produce several polynomiographs that are interesting from the artistic and computational points of view.

The rest of the paper is organised as follows. In Section 2, we introduce the basic family of iterations and some iterative methods from the fixed point theory and the RK iteration process. In Section 3, we present some escape criterion results based on the RK iteration. In Section 4, we present our algorithm and some examples of polynomiographs using RK iteration. Section 5 concludes the paper and shows some future directions.

2. Basic Family of Iterations and Other Iterative Methods

Given a polynomial $p\in \mathbb{C}\left[\mathbb{Z}\right]$ of degree $n\ge 2,$ for each $m\ge 2,$ we define a sequence of functions $D_m:\mathbb{C}\to \mathbb{C}$ such that $D_0\left(z\right)=1$ as follows:

$$D_m\left(z\right)=det\left(\begin{array}{ccccc}{p}^{\prime }\left(z\right)& \frac{p^{″}\left(z\right)}{2!}& \cdots & \frac{p^{(m-1)}\left(z\right)}{(m-1)!}& \frac{p^{\left(m\right)}\left(z\right)}{m!}\\ p\left(z\right)& p^{\prime }\left(z\right)& \cdots & \cdots & \frac{p^{(m-1)}\left(z\right)}{(m-1)!}\\ ⋮& ⋮& \cdots & ⋮& ⋮\\ 0& p\left(z\right)& \cdots & p^{\prime }\left(z\right)& \frac{p^{″}\left(z\right)}{2!}\\ 0& 0& \cdots & p\left(z\right)& p^{\prime }\left(z\right)\end{array}\right).$$

Then, the basic family of iterations is given by

$$B_m\left(z\right)=z-p\left(z\right)\frac{D_{m-2}\left(z\right)}{D_{m-1}\left(z\right)},m=2,3,4,\cdots .$$

(2)

The basic family of iterations was introduced by Kalantari [2] as a generalisation of several methods for finding the roots of polynomials. The iterative behaviour of individual members of the basic family was studied in [2]. It is easy to see that the first three members of the basic family are:

$$\begin{array}{ccc}\hfill B_2\left(z\right)& =& z-\frac{p\left(z\right)}{p^{\prime }\left(z\right)},\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill B_3\left(z\right)& =& z-\frac{2p^{\prime }\left(z\right)p\left(z\right)}{2p^{\prime }{\left(z\right)}^2-p^{″}\left(z\right)p\left(z\right)},\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill B_4\left(z\right)& =& z-\frac{6p^{\prime }{\left(z\right)}^{2}p\left(z\right)-3p^{″}\left(z\right)p{\left(z\right)}^2}{p^{‴}\left(z\right)p{\left(z\right)}^2+6p^{\prime }{\left(z\right)}^3-6p^{″}\left(z\right)p^{\prime }\left(z\right)p\left(z\right)}.\hfill \end{array}$$

(5)

We see that $B_2$ is the same as in Newton’s method for complex polynomials and $B_3$ is the same as in Halley’s method for finding the roots of polynomials; see [7]. Moreover, the basic family has several fundamental properties. It is closely related to a non-trivial determinant generalisation of Taylor’s theorem [18]. For more information on the basic family, such as the multipoint version and the order of convergence, see [18,19]. Further, a detailed list of the theoretical and computation properties of the basic family can be found in, for instance, refs. [18,19,20,21].

Let $(X,d)$ be a complete metric space and $T:X\to X$ be a mapping. A point $x\in X$ is called a fixed point T if $Tx=x.$ Many iterative methods have been introduced for approximating the fixed points of several mappings in the literature. We recall some well-known methods for any given $x_0\in X$ as follows:

(i)

The standard Picard iteration [22] is defined by

$$x_{n+1}=T{x}_n,n=0,1,2,\cdots ,$$

(6)

(ii)

The Mann iteration [23] is given by

$$x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n,n=0,1,2,\cdots ,$$

where ${\alpha }_n\in (0,1]$.

(iii)

The Ishikawa iteration [24], which is a two-step process, is given by

$$\left\{\begin{array}{c}{x}_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_{n}T{y}_n,\hfill \\ y_n=(1-{\beta }_n)x_n+{\beta }_{n}T{x}_n,n=0,1,2,\cdots ,\hfill \end{array}\right.$$

where ${\alpha }_n\in (0,1]$ and ${\beta }_n\in [0,1].$

(iv)

The Picard–Mann iteration of Khan [25] is given as

$$\left\{\begin{array}{c}{x}_{n+1}=T{y}_n\hfill \\ y_n=(1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n,n=0,1,2,\cdots ,\hfill \end{array}\right.$$

where ${\alpha }_n\in (0,1].$

(v)

The S-iteration of Agarwal [26] is defined by

$$\left\{\begin{array}{c}{x}_{n+1}=(1-{\alpha }_n)T{x}_n+{\alpha }_{n}T{y}_n,\hfill \\ y_n=(1-{\beta }_n)x_n+{\beta }_{n}T{x}_n,n=0,1,2,\cdots ,\hfill \end{array}\right.$$

where ${\alpha }_n\in (0,1]$ and ${\beta }_n\in [0,1].$

The relationship between the iterative methods can be found in, for instance, ref. [12]. Moreover, Gdawiec et al. [12,14] have used different iteration methods, different convergence tests, and different colouring methods to obtain a variety of polynomiographs.

Recently, Ritika and Khan [17] introduced the RK iteration as a generalisation of many known iterations. They also showed that the RK converges faster than most of the known iteration processes for approximating the fixed point of a mapping. The RK iteration is a four-step process defined as follows. Given $x_0\in X$,

$$\left\{\begin{array}{c}{x}_{n+1}=T{w}_n,\hfill \\ w_n=T((1-{\alpha }_n)y_n+{\alpha }_{n}T{y}_n),\hfill \\ y_n=T((1-{\beta }_n)z_n+{\beta }_{n}T{z}_n),\hfill \\ z_n=T((1-{\gamma }_n)x_n+{\gamma }_{n}T{x}_n),n=0,1,2,\cdots ,\hfill \end{array}\right.$$

(7)

where ${\alpha }_n\in (0,1]$ and ${\beta }_n,{\gamma }_n\in [0,1].$ Ritika and Khan [17] proved that the sequence generated by the RK iteration convergences to the fixed point of a generalised nonexpansive mapping in a CAT(0) space (see [17] for details). In this paper, we also introduce a generalisation of the RK iteration using the s-convex combination as follows. Given $x_0\in X$,

$$\left\{\begin{array}{c}{x}_{n+1}=T{w}_n,\hfill \\ w_n=T({(1-{\alpha }_n)}^s{y}_n+{\alpha }_n^{s}T{y}_n),\hfill \\ y_n=T({(1-{\beta }_n)}^s{z}_n+{\beta }_n^{s}T{z}_n),\hfill \\ z_n=T({(1-{\gamma }_n)}^s{x}_n+{\gamma }_n^{s}T{x}_n),n=0,1,2,\cdots ,\hfill \end{array}\right.$$

(8)

where ${\alpha }_n\in (0,1],$ ${\beta }_n, {\gamma }_n\in [0,1]$ and $s\in (0,1].$ Clearly, if $s=1$, we obtain the RK iteration (7).

3. Escape Criterion Results

The escape criterion plays a vital role in the generation and analysis of Julia sets, Mandelbrot sets, and their generalisations. In this section, we describe some escape criteria for general polynomials of the form

$$Q_r^n\left(x\right)=x^n+q{x}^2+mx+r,$$

(9)

where $n\ge 2,$ $m, r\in \mathbb{C},$ $q\in \{0,1\}.$

Following from [11], it is easy to see that (9) contains quadratic, cubic, and higher-order polynomials. To explain this, the presence of the quadratic term $q{x}^2$ makes it more general. When $q=0,$ we obtain the polynomial $Q_r^n\left(x\right)=x^n+mx+r$, which has been studied by some authors—for instance, Abbas et al. [27] and Nazeer et al. [28]. Moreover, complex polynomials of the form (9) can be found in several problems arising from engineering, such as digital signal processing. In particular, they are used in determining the pole-zero plots for signals and studying the structure and solutions of linear-time-variant problems [29].

We now give our escape criterion result for general complex polynomials. In the sequel, without loss of generalisation, we denote ${\alpha }_n=\alpha ,{\beta }_n=\beta ,{\gamma }_n=\gamma .$

Theorem 1.

Let $Q_r^n\left(x\right)=x^n+q{x}^2+mx+r,$ and suppose

$$\left|x\right|\ge \left|r\right|>max\left\{\frac{2(1+|m\left|\right)}{\gamma \left(\right|r^{n-2}|+q)},\frac{2(1+|m\left|\right)}{\beta \left(\right|r^{n-2}|+q)},\frac{2(1+|m\left|\right)}{\alpha \left(\right|r^{n-2}|+q)}\right\},$$

where $\alpha ,\beta ,\gamma \in (0,1],$ $r,m\in \mathbb{C}$ and $q\in \{0,1\}.$ Define $\left\{x_k\right\}$ as in (7). Then, $|x_k|\to \infty$ as $k\to \infty .$

Proof. 

Put $x_0=x,w_0=w,y_0=y,z_0=z$ and $u=(1-\gamma )x+\gamma Q_{r}x.$ Then,

$$\begin{array}{ccc}\hfill \left|u\right|& =& |(1-\gamma )x+\gamma Q_r^{n}x|\hfill \\ & =& |(1-\gamma )x+\gamma (x^n+q{x}^2+mx+r)|\hfill \\ & \ge & |(1-\gamma )x+\gamma (x^n+q{x}^2+mx)|-\gamma |r|\hfill \\ & \ge & |(1-\gamma )x+\gamma (x^n+q{x}^2+mx)|-\gamma |x|\hfill \\ & \ge & \gamma |x^n+q{x}^2|-(1-\gamma +\gamma \left|m\right|\left)\right|x|-\gamma |x|\hfill \\ & =& \gamma |x^n+q{x}^2|-(1+\gamma \left|m\right|\left)\right|x|\hfill \\ & =& \left|x\right|\left(\gamma \left|x\right|\left(\right|x^{n-2}|+q)-(1+\gamma |m\left|\right)\right).\hfill \end{array}$$

Moreover, since $\gamma \in (0,1],$ we obtain

$$\begin{array}{ccc}\hfill \left|u\right|& \ge & \left|x\right|\left(\gamma \left|x\right|\left(\right|x^{n-2}|+q)-(1+|m\left|\right)\right)\hfill \\ & \ge & \left|x\right|\left(\gamma \left|x\right|\left(\right|r^{n-2}|+q)-(1+|m\left|\right)\right)\hfill \\ & =& \left|x\right|(1+\left|m\right|)\left(\frac{\gamma \left|x\right|\left(\right|r^{n-2}|+q)}{1+\left|m\right|}-1\right).\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill \left|u\right|& \ge & \frac{\left|u\right|}{1+\left|m\right|}\hfill \\ & \ge & \left|x\right|\left(\frac{\gamma \left|x\right|\left(\right|r^{n-2}|+q)}{1+\left|m\right|}-1\right).\hfill \end{array}$$

From the hypothesis of our theorem, we know that

$$\left|x\right|>\frac{2(1+|m\left|\right)}{\gamma \left(\right|r^{n-2}|+q)},$$

which implies that

$$\frac{\gamma \left|x\right|\left(\right|r^{n-2}|+q)}{1+\left|m\right|}-1>1.$$

(10)

Hence,

$$\left|u\right|>\left|x\right|.$$

Consequently,

$$\begin{array}{ccc}\hfill \left|z\right|& =& |Q_r^n\left(u\right)|\hfill \\ & =& |u^n+q{u}^2+mu+r|\hfill \\ & \ge & |u^n+q{u}^2+mu|-|r|\hfill \\ & \ge & |u^n+q{u}^2+mu|-|u|.\hfill \end{array}$$

Moreover, since $\gamma \in (0,1],$ we have

$$\begin{array}{ccc}\hfill \left|z\right|& \ge & \gamma |u^n+q{u}^2|-|m\left|\right|u|-|u|\hfill \\ & =& \gamma |u^2\left|\right(|u^{n-2}|+q)-(1+|m\left|\right)\left|u\right|\hfill \\ & =& \left|u\right|\left(\gamma \left|u\right|\left(\right|u^{n-2}|+q)-(1+|m\left|\right)\right)\hfill \\ & >& \left|u\right|\left(\gamma \left|u\right|\left(r^{n-2}\right|+q)-(1+|m\left|\right)\right).\hfill \end{array}$$

Since $\left|u\right|>\left|x\right|,$ thus, we have

$$\begin{array}{ccc}\hfill \left|z\right|& \ge & \left|x\right|\left(\gamma \left|x\right|\left(\right|r^{n-2}|+q)-(1+|m\left|\right)\right)\hfill \\ & =& \left|x\right|\left(\frac{\gamma \left|x\right|\left(\right|r^{n-2}|+q)}{1+\left|m\right|}-1\right).\hfill \end{array}$$

From (10), we obtain

$$\left|z\right|\ge \left|x\right|.$$

Following a similar argument as above, we have

$$\left|y\right|>\left|z\right|\mathrm{and}\left|w\right|>\left|y\right|,$$

$$\left|w\right|>\left|y\right|>\left|z\right|>\left|x\right|.$$

Then, from (7), we have

$$\begin{array}{ccc}\hfill |x_1|& =& |Q_r^n\left(w\right)|\hfill \\ & =& |w^n+q{w}^2+mw+r|\hfill \\ & \ge & |w^n+q{w}^2+mw|-|r|\hfill \\ & \ge & |w^n+q{w}^2+mw|-|w|\hfill \\ & \ge & \alpha |w^n+q{w}^2|-|m\left|\right|w|-|w|\hfill \\ & =& \alpha |w^n+q{w}^2|-(1+\left|m\right|\left)\right|w|\hfill \\ & =& \left|w\right|\left(\alpha \left|w\right|\left(\right|w^{n-2}|+q)-(1+|m\left|\right)\right)\hfill \\ & \ge & \left|x\right|\left(\alpha \left|x\right|\left(\right|x^{n-2}|+q)-(1+|m\left|\right)\right)\hfill \end{array}$$

(11)

Hence,

$$\begin{array}{ccc}\hfill |x_1|& \ge & \left|x\right|\left(\alpha \left|x\right|\left(\right|r^{n-2}|+q)-(1+|m\left|\right)\right)\hfill \\ & =& \left|x\right|\left(\frac{\alpha \left|x\right|\left(\right|r^{n-2}|+q)}{1+\left|m\right|}-1\right).\hfill \end{array}$$

From the hypothesis of the theorem, we have

$$\left|x\right|>\frac{2(1+|m\left|\right)}{\alpha \left(\right|r^{n-2}|+q)}.$$

This implies that

$$\frac{\alpha \left|x\right|\left(\right|r^{n-2}|+q)}{1+\left|m\right|}-1>1.$$

Hence, there exists a real number $\rho >0$ such that

$$\frac{\alpha \left|x\right|\left(\right|r^{n-2}|+q)}{1+\left|m\right|}-1>1+\rho .$$

Therefore,

$$\begin{array}{c}\hfill |x_1|>(1+\rho )|x|.\end{array}$$

Hence, by induction, we obtain

$$|x_n{|>(1+\rho )}^n\left|x\right|.$$

This means that $|x_n|\to \infty$ as $n\to \infty .$ This completes the proof.  □

As a consequence, we obtain the escape criterion result for (8).

Theorem 2.

Let $\left\{x_n\right\}$ be as defined in (8) and $Q_r^n\left(x\right)=x^n+q{x}^2+mx+r.$ Suppose

$$\left|x\right|\ge \left|r\right|>max\left\{\frac{2(1+|m\left|\right)}{s\gamma \left(\right|r^{n-2}|+q)},\frac{2(1+|m\left|\right)}{s\beta \left(\right|r^{n-2}|+q)},\frac{2(1+|m\left|\right)}{s\alpha \left(\right|r^{n-2}|+q)}\right\},$$

(12)

where $s\in (0,1),$ $\alpha ,\beta ,\gamma \in (0,1],$ $r,m\in \mathbb{C}$ and $q\in \{0,1\}.$ Then, $|x_n|\to \infty$ as $n\to \infty .$

Proof. 

Following a similar process as in the proof of Theorem 1, from (8), we obtain

$$\begin{array}{ccc}\hfill \left|u\right|& =& {|(1-\gamma )}^{s}x+{\gamma }^s{Q}_r^{n}x|\hfill \\ & =& {|(1-\gamma )}^{s}x+{\gamma }^s(x^n+q{x}^2+mx+r)|\hfill \\ & =& {|(1-\gamma )}^{s}x+{(1-(1-\gamma ))}^s(x^n+q{x}^2+mx+r)|.\hfill \end{array}$$

By the binomial expression up to linear terms of $\gamma$ and $(1-\gamma ),$ we obtain

$$\begin{array}{ccc}\hfill \left|u\right|& \ge & {|(1-\gamma )}^{s}x+(1-s(1-\gamma ))(x^n+q{x}^2+mx+r)|\hfill \\ & \ge & |(1-s\gamma )x+s\gamma (x^n+q{x}^2+mx)+r|(\mathrm{because}1-s(1-\gamma )\ge s\gamma )\hfill \\ & \ge & |(1-s\gamma )x+s\gamma (x^n+q{x}^2+mx)|-s\gamma |r|\hfill \\ & \ge & |(1-s\gamma )x+s\gamma (x^n+q{x}^2+mx)|-s\gamma |x|\hfill \\ & \ge & s\gamma |x^n+q{x}^2|-(1-s\gamma +s\gamma \left|m\right|\left)\right|x|-s\gamma |x|\hfill \\ & =& s\gamma |x^n+q{x}^2|-(1+s\gamma \left|m\right|\left)\right|x|\hfill \\ & =& \left|x\right|\left(s\gamma \left|x\right|\left(\right|x^{n-2}|+q)-(1+s\gamma |m\left|\right)\right).\hfill \end{array}$$

(13)

Since $s\in (0,1)$ and $\gamma \in (0,1],$ then $s\gamma \in (0,1),$ and hence

$$\begin{array}{ccc}\hfill \left|u\right|& \ge & \left|x\right|\left(s\gamma \left|x\right|\left(\right|x^{n-2}|+q)-(1+|m\left|\right)\right)\hfill \\ & \ge & \left|x\right|\left(s\gamma \left|x\right|\left(\right|r^{n-2}|+q)-(1+|m\left|\right)\right)\hfill \\ & =& \left|x\right|(1+\left|m\right|)\left(\frac{s\gamma \left|x\right|\left(\right|r^{n-2}|+q)}{1+\left|m\right|}-1\right).\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill \left|u\right|& \ge & \frac{\left|u\right|}{1+\left|m\right|}\hfill \\ & \ge & \left|x\right|\left(\frac{s\gamma \left|x\right|\left(\right|r^{n-2}|+q)}{1+\left|m\right|}-1\right).\hfill \end{array}$$

Using condition (12), we obtain

$$\frac{s\gamma \left|x\right|\left(\right|r^{n-2}|+q)}{1+\left|m\right|}-1>1.$$

Hence,

$$\left|u\right|>\left|x\right|.$$

Then,

$$\begin{array}{ccc}\hfill \left|z\right|& =& |Q_r^n\left(u\right)|\hfill \\ & =& |u^n+q{u}^2+mu+r|\hfill \\ & \ge & |u^n+q{u}^2+mu|-|r|\hfill \\ & \ge & |u^n+q{u}^2+mu|-|u|.\hfill \end{array}$$

Since $s\gamma \in (0,1),$ we have

$$\begin{array}{ccc}\hfill \left|z\right|& \ge & s\gamma |u^n+q{u}^2|-|m\left|\right|u|-|u|\hfill \\ & =& s\gamma |u^2\left|\right(|u^{n-2}|+q)-(1+|m\left|\right)\left|u\right|\hfill \\ & =& \left|u\right|\left(s\gamma \left|u\right|\left(\right|u^{n-2}|+q)-(1+|m\left|\right)\right)\hfill \\ & >& \left|u\right|\left(s\gamma \left|u\right|\left(r^{n-2}\right|+q)-(1+|m\left|\right)\right).\hfill \end{array}$$

Since $\left|u\right|>\left|x\right|,$ thus, we have

$$\begin{array}{ccc}\hfill \left|z\right|& \ge & \left|x\right|\left(s\gamma \left|x\right|\left(\right|r^{n-2}|+q)-(1+|m\left|\right)\right)\hfill \\ & =& \left|x\right|\left(\frac{s\gamma \left|x\right|\left(\right|r^{n-2}|+q)}{1+\left|m\right|}-1\right).\hfill \end{array}$$

Thus, we obtain

$$\left|z\right|\ge \left|x\right|.$$

Following a similar argument, we obtain

$$\left|y\right|>\left|z\right|\mathrm{and}\left|w\right|>\left|y\right|,$$

$$\left|w\right|>\left|y\right|>\left|z\right|>\left|x\right|.$$

Hence, from (8), we have

$$\begin{array}{ccc}\hfill |x_1|& =& |Q_r^n\left(w\right)|\hfill \\ & =& |w^n+q{w}^2+mw+r|\hfill \\ & \ge & |w^n+q{w}^2+mw|-|r|\hfill \\ & \ge & |w^n+q{w}^2+mw|-|w|\hfill \\ & \ge & s\alpha |w^n+q{w}^2|-|m\left|\right|w|-|w|\hfill \\ & =& s\alpha |w^n+q{w}^2|-(1+\left|m\right|\left)\right|w|\hfill \\ & =& \left|w\right|\left(s\alpha \left|w\right|\left(\right|w^{n-2}|+q)-(1+|m\left|\right)\right)\hfill \\ & \ge & \left|x\right|\left(s\alpha \left|x\right|\left(\right|x^{n-2}|+q)-(1+|m\left|\right)\right)\hfill \end{array}$$

(14)

Hence,

$$\begin{array}{ccc}\hfill |x_1|& \ge & \left|x\right|\left(s\alpha \left|x\right|\left(\right|r^{n-2}|+q)-(1+|m\left|\right)\right)\hfill \\ & =& \left|x\right|\left(\frac{s\alpha \left|x\right|\left(\right|r^{n-2}|+q)}{1+\left|m\right|}-1\right).\hfill \end{array}$$

From the hypothesis of the theorem, we have

$$\left|x\right|>\frac{2(1+|m\left|\right)}{s\alpha \left(\right|r^{n-2}|+q)}.$$

This implies that

$$\frac{s\alpha \left|x\right|\left(\right|r^{n-2}|+q)}{1+\left|m\right|}-1>1.$$

Hence, there exists a real number $\lambda >0$ such that

$$\frac{s\alpha \left|x\right|\left(\right|r^{n-2}|+q)}{1+\left|m\right|}-1>1+\lambda .$$

Therefore,

$$\begin{array}{c}\hfill |x_1|>(1+\lambda )|x|.\end{array}$$

Hence, by induction, we obtain

$$|x_n{|>(1+\lambda )}^n\left|x\right|.$$

This means that $|x_n|\to \infty$ as $n\to \infty .$ This completes the proof.  □

4. Polynomiograph Generation

4.1. Algorithms

For generating the polynomiographs, we choose a polynomial $p\in \mathbb{C}\left[\mathbb{Z}\right],$ the parameters $\alpha ,\beta ,\gamma ,s$ for the RK iteration, the area $A\in \mathbb{C}$ for the polynomiograph, the maximum number of iterations $M\in \mathbb{N}$, and the accuracy $ϵ>0$ for the following convergence test:

$$|x_{n+1}-x_n|<ϵ.$$

(15)

When the convergence test (15) is satisfied, we determine the colour for the point in the area A using the method of number of iterations (see [12]). The pseudo-code for generating the polynomiographs is given in Algorithm 1.

Algorithm 1 RK algorithm for generating polynomiography.

Input: Choose $$\begin{array}{c}\hfill \begin{array}{ccc}p\in \mathbb{C}\left[\mathbb{Z}\right]& --& \mathrm{polynomial}\mathrm{of}\mathrm{degree}\ge 2,\\ x_0\in \mathbb{C}& --& \mathrm{starting}\mathrm{point},\\ \alpha \in (0,1],\beta \in [0,1],\gamma \in [0,1]& --& \mathrm{parameters}\mathrm{of}\mathrm{mixed}\mathrm{iteration},\\ s\in (0,1]& --& \mathrm{parameter}\mathrm{for}\mathrm{s}-\mathrm{convexity},\\ M& --& \mathrm{maximum}\mathrm{number}\mathrm{of}\mathrm{iteration},\\ A\in \mathbb{C}& --& \mathrm{area}\mathrm{for}\mathrm{the}\mathrm{polynomiographs}.\end{array}\end{array}$$ (16) While:$n\le M$, compute $$\left\{\begin{array}{c}{z}_n=B_m({(1-\gamma )}^s{x}_n+{\gamma }^s{B}_m{x}_n)\hfill \\ y_n=B_m({(1-\beta )}^s{z}_n+{\beta }^s{B}_m{z}_n)\hfill \\ w_n=B_m({(1-\alpha )}^s{y}_n+{\alpha }^s{B}_m{y}_n)\hfill \\ x_{n+1}=B_m{w}_n\hfill \end{array}\right.$$ (17) Stopping criterion:$|x_{n+1}-x_n|<ϵ$. Output: Colour c of $x_0.$

4.2. Visualisation of Polynomiographs

In this section, some examples of polynomiographs are presented using the modifications of the RK iterations with the basic family of iterations. The pseudo-code for generating the polynomiographs is presented in Algorithm 1. The experiments were carried out using MATLAB programming on a Lenovo PC with specification Intel(R) Core $i7$-5600u, @ 3.2 GHz processor, 4.00 GB RAM, Window 10(64-bit operating system). The resolution of all graphics was set as $600\times 600$ pixels, the maximum number of iterations is chosen as $M=300$, and the accuracy is set as $ϵ=0.0001.$ Moreover, the area of the images is set as ${[-2,2]}^2.$

Example 1.

In this example, we present some polynomiographs generated by by the RK iteration, RK iteration with s-convexity, Mann iteration, Ishikawa iteration, S-iteration, and Picard–Mann iteration. We choose ${\alpha }_k=\frac{1}{k+1},{\beta }_k=\frac{2k}{3k+1}$ and ${\gamma }_k=\frac{1}{10(k+1)}$, and $s=0.65$ for the s-convexity. We choose the following polynomials:

$$\left(a\right)p\left(z\right)=z^6-1\mathrm{and}\left(b\right)p\left(z\right)=z^{11}+z^8-z^7+2z^6+3z^5+3z^4-2z^3-1.$$

The polynomiographs generated by the methods are shown in Figure 1 and Figure 2.

As a remark, we see from Figure 1 that the RK iteration is efficient for generating polynomiographs. Moreover, the RK iteration with s-convexity generates a distinct polynomiograph where the twisting in the image can be explained as the effect of the s-convexity on the polynomiograph. This was also observed in the polynomiographs obtained by other authors; see, for instance [16]. Moreover, as seen in Figure 2, the RK iteration generates a unique polynomiograph from another iteration. The s-convexity in this case produces a more interesting image than the other iterations.

Example 2.

In this example, we study the effects of the s-convexity value on the visual polynomiographs generated by Algorithm 1. In particular, we take ${\alpha }_k=\frac{1}{k+1},{\beta }_k=0.45,{\gamma }_k=0.8$, and $p\left(z\right)=z^5+2z^4+2z^3-3z^2-1.$ We choose the following values for the s-convexity:

$$\left(a\right)s=1,\left(b\right)s=0.95\left(c\right)s=0.8\left(d\right)s=0.5\left(e\right)s=0.25\left(f\right)s=0.05.$$

The polynomiographs generated by the methods are shown in Figure 3.

It is observed that as the value of s decreases, the iteration loses the efficiency of producing an iterating polynomiograph. This is expected because as the value of s decreases, the iterations approach very closely to zero, which produces the same colour at each point.

Example 3.

In this example, we investigate the effects of the control sequence $\left\{{\alpha }_k\right\},\left\{{\beta }_k\right\}$, and $\left\{{\gamma }_k\right\}$ on the polynomiographs generated by the RK iteration. We consider $p\left(z\right)=z^4+1$ for the RK iteration without s-convexity, with the following values of the control sequence:

Case I: 

${\alpha }_k=0.39;{\beta }_k=0.25;{\gamma }_k=0.95;$

Case II: 

${\alpha }_k=1/(k+1);{\beta }_k=1/\sqrt{(}k+1);{\gamma }_k=|sin(k+1)|;$

Case III: 

${\alpha }_k=1/7^k;{\beta }_k=1/\left(5^k\right);{\gamma }_k=1/\left(2^k\right);$

Case IV: 

${\alpha }_k=0.99;{\beta }_k=1;{\gamma }_k=1;$

Case V: 

${\alpha }_k=0.7;{\beta }_k=0.39;{\gamma }_k=1;$

Case VI: 

${\alpha }_k=0.07;{\beta }_k=0.039;{\gamma }_k=0.01;$

The polynomiographs generated are shown in Figure 4. It is important to mention that there are no special requirements for choosing these sequences apart from the condition that $\left\{{\alpha }_k\right\}\subset (0,1)$ and $\left\{{\beta }_k\right\},\left\{{\gamma }_k\right\}\subset [0,1].$ Hence, these sequences are chosen arbitrarily. Furthermore, we take $p\left(z\right)=5z^{10}+3z^8+z^2+3$ for the RK iteration with s-convexity. The polynomiographs generated are shown in Figure 5.

We observe that there is no significant effect of the control sequence on the polynomiographs obtained using RK iteration without s-convexity, although there is a slight change in colour at each case. However, for the RK iteration with s-convexity, we observe a significant effect on the obtained polynomiographs.

5. Conclusions

In this paper, we replaced the standard Picard iteration with a new four-step RK iteration in the root-finding methods from the basic family of iterations. We also introduced a generalisation of the RK iteration using the s-convex combination method. Further, we presented some polynomiograph examples showing the effects of the new processes on the dynamics, colours, and shapes of the polynomiographs. We infer from Experiments 1–6 that the lower the value of s in s-convexity, the higher the number of iterations required by the iteration process to approximate the roots of the polynomials. Moreover, the computational results show that the value of s in the s-convexity has a great impact on the time taken by the iteration for approximating the roots of the polynomials.

The results of the paper can be modified in many directions, e.g., by using different convergence tests of different colouring methods, as in, e.g., [12]. Moreover, we can modify other families of iterations, such as the Euler–Schöder family, with RK iteration and its s-convexity. Furthermore, we can check the modification of the RK iterations with systems of polynomial equations. The results obtained in the processes can be applied to computer graphics where aesthetic patterns can be used, such as texture generation, animation, and tapestry design.