DYNAMICS OF MODIFIED ABBASBANDY'S METHOD FOR SOLVING ...

International Journal of Pure and Applied Mathematics Volume 111 No. 3 2016, 507-523 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.12732/ijpam.v111i3.13

AP ijpam.eu

DYNAMICS OF MODIFIED ABBASBANDY’S METHOD FOR SOLVING NONLINEAR EQUATIONS Waqas Nazeer1 , Shah Hussain2 , Irum Sarfraz3 , Chahn Yong Jung4 , Shin Min Kang5 1 Division

of Science and Technology University of Education Lahore, 54000, PAKISTAN 2 Department of Mathematics Minhaj University Lahore, 54000, PAKISTAN 3 Department of Mathematics Lahore Leads University Lahore, 54810, PAKISTAN 4 Department of Business Administration Gyeongsang National University Jinju, 52828, KOREA 5 Department of Mathematics and RINS Gyeongsang National University Jinju, 52828, KOREA

Abstract:

In this note we analyzed an iterative method for nonlinear equations with con-

vergence of order six. Some text examples was solved to check validity and efficiency of this iterative method. Moreover we visualized polynomiography of some complex polynomials via this iterative method. Received:

October 11, 2016

Revised:

December 1, 2016

Published:

December 19, 2016

c 2016 Academic Publications, Ltd.

url: www.acadpubl.eu

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AMS Subject Classification: 65H05, 65D32 Key Words:

nonlinear equation, iterative method, Newton method, Halley’s method,

Househ¨ older’s method, Abbasbandy’s method, Noor and Noor method, modified Abbasbandy’s method, polynomiography

1. Introduction The boundary value problems in Kinetic theory of gases, elasticity and other applied areas are mostly reduced in solving single variable nonlinear equations. Hence, the problem of approximating a solution of the nonlinear equations f (x) = 0, is important. The numerical methods for the roots of such equations are called iterative methods [29]. Many such iterative methods for solving nonlinear equations are in literature for example [29, 28, 9, 21, 1, 30, 10, 26, 27, 6, 4, 5, 7, 8, 11, 19, 20, 2, 3, 16, 17, 22, 23] and the reference therein. There are two types of iterative methods, i.e. derivative free methods [28] and, higher order iterative methods involving derivatives [9, 21, 1, 30, 10, 26, 27, 6, 4, 5, 7, 8, 11, 19, 20, 2, 3, 16, 17, 22]. Here, we are interested in finding higher order iterative method involving derivative. The concept of polynomiography arose out of a problem concerning polynomial root-finding by Kalantari [14, 12, 13]. Kalantari became interested in polynomial root-finding when he was designing masters questions for an exam, related to approximations of the square root of two. As he delved deeper into the matter, his interest was piqued at the possibility of generating computer visualizations of the root-finding process. Images thus generated would be related to fractals, but distinct in that the former affords more control over the design than the latter. Additionally, a polynomiography image, called a polynomiograph, does not necessarily exhibit fractal patterns. Following figures are well known two polynomiographs. The name polynomiography was subsequently coined in 2000 as a portmanteau of the words polynomial and the Greek suffix-graphy. In 2005 Kalantari [13] obtained a patent for the technology of polynomiography. Definition 1.1. In its simplest form, a polynomial is a mathematical expression involving variables of varying powers such as x2 − 3x + 4.

A naive way to find the roots of a polynomial expression is to graph the equation and find the zeroes, where the graph crosses the horizontal (x-)axis. For more complex polynomials, finding the roots graphically can be quite difficult. In light of this, mathematicians have developed algorithms for finding the roots of polynomials without having to plot them graphically. These are

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Figure A. Non-fractal polynomiograph of the polynomial p(z) = z 3 − 1

Figure B. Newton fractal for three degree-3 roots p(z) = z 3 − 1 colored by root reached

usually methods that use iteration to converge upon a root; they locate an area that is a starting point for the root of the polynomial, then progress using an algorithm to find a better point, in order to reach a limit for the approximation of the root. Polynomiography is the algorithmic visualization of polynomial equations, making use of one or more of these iterative techniques. The iterative methods may consist of using iteration functions. A particular family of iterations used extensively in polynomiography is the Basic Family. Polynomiography results in a 2D image called a polynomiograph that may or may not exhibit fractal or chaotic behavior. More broadly, a polynomiograph may be a 3D or even 4D object. In this paper, we proposed and analyzed a new iterative method with con-

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vergence of order six. Polynomiographs of some complex polynomials is visualized via this iterative method.

2. Iterative Methods Consider the nonlinear algebraic equation f (x) = 0.

(2.1)

We assume that α is a simple zero of Eq. (2.1) and γ is an initial guess sufficiently close to α. Using the Taylor’s series around γ for Eq. (2.1), we have f (γ) + (x − γ)f ′ γ) +

1 (x − γ)2 f ′′ (γ) + · · · = 0. 2!

(2.2)

If f ′ (γ) 6= 0, we can evaluate the above expression as follows: f (xk ) + (x − xk )f ′ (xk ) = 0. This formulation is used to suggest the following iterative method. Algorithm 2.1. For a given x0 , compute the approximate solution xn+1 by the iterative scheme f (xn ) xn+1 = xn − ′ . (2.3) f (xn ) This is well known the Newton’s method (NM) for root-finding of nonlinear functions, which converges quadratically [29, 5]. Also from (2.2), we obtain x=γ−

2f (γ)f ′ (γ) . 2f ′2 (γ) − f (γ)f ′′ (γ)

This formulation allows us to suggest the following iterative method for solving nonlinear equation (2.1). Algorithm 2.2. For a given x0 , compute the approximate solution xn+1 by the iterative scheme xn+1 = xn −

2f (xn )f ′ (xn ) . 2f ′2 (xn ) − f (xn )f ′′ (xn )

This is known as the Halley’s Method, which has cubic convergence[29, 9, 21, 6, 5].

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Algorithm 2.3. For a given x0 , compute the approximate solution xn+1 by the iterative scheme xn+1 = xn −

f 2 (xn )f ′′ (xn ) f (xn ) − . f ′ (xn ) 2f ′3 (xn )

This is so-called the Househ¨ older method, which has convergence of order three[29, 5]. Abbasbandy [1] improve the Newton-Raphson method by the modified Adomian decomposition method, and develop following third order iterative method. Algorithm 2.4. For a given x0 , compute the approximate solution xn+1 by the iterative scheme xn+1 = xn −

f (xn ) f 2 (xn )f ′′ (xn ) f 3 (xn )f ′′′ (xn ) − − . f ′ (xn ) 2f ′3 (xn ) 6f ′4 (xn )

This is so-called the Abbasbandy’s method for root-finding of nonlinear functions. Noor and Noor [25] suggested the following two-step method Algorithm 2.5. For a given x0 , compute the approximate solution xn+1 by the iterative scheme f (xn ) , f ′ (xn ) 2f (yn )f ′ (yn ) . = yn − ′2 2f (yn ) − f (yn )f ′′ (yn )

y n = xn − xn+1

For more details, see [24, 15] and references therein. Now we propose the following algorithm. Algorithm 2.6. For a given x0 , compute the approximate solution xn+1 by the following iterative schemes: f (xn ) , f ′ (xn ) f 2 (yn )f ′′ (yn ) f 3 (yn )f ′′′ (yn ) f (yn ) − − . = yn − ′ f (yn ) 2f ′3 (yn ) 6f ′4 (yn )

y n = xn − xn+1

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3. Convergence Analysis In this section, we will show that the convergence order of Algorithm 2.6 is at least six. Theorem 3.1. Let α be a root of the equation f (x) = 0. If f (x) is sufficiently smooth in the neighborhood of α, then Algorithm 2.6 has at least sixth-order convergence. Proof. To prove the convergence of Algorithm 2.6 is six, suppose that α is a root of the equation f (x) = 0 and en be the error at n-th iteration, than en = xn − α then by using Taylor series expansion, we have 1 1 1 ′′ f (xn )e2n + f ′′′ (xn )e3n + f (iv) (xn )e4n 2! 3! 4! 1 1 + f (v) (xn )e5n + f (vi) (xn )e6n + O(e7n ), 5! 6!

f (xn ) = f ′ (xn )en +

f (xn ) = f ′ (α)[en + c2 e2n + c3 e3n + c4 e4n + c5 e5n + c6 e6n + c7 e7n + O(e8n )], f ′ (xn ) = f ′ (α)[1 + 2c2 en + 3c3 e2n + 4c4 e3n + 5c5 e4n + 6c6 e5n + 7c7 e6n + O(e7n )],

(3.1) (3.2)

where

1 f (n) (α) . n! f ′ (α) With the help of (3.1) and (3.2), we get cn =

yn = f ′ (α)[α + c2 e2n + (2c3 − 2c22 )e3n + (3c4 − 7c2 c3 + 4c32 )e4n + (−6c23 + 20c3 c22 − 10c2 c4 + 4c5 − 8c42 )e5n + (−17c4 c3

+ 28c4 c22 − 13c2 c5 + 5c6 + 33c2 c23 − 52c3 c32 + 16c52 )e6n

(3.3)

+ (−22c5 c3 + 36c5 c22 + 6c7 − 16c2 c6 − 12c24 + 92c4 c2 c3

− 72c4 c32 + 18c33 − 126c23 c22 + 128c3 c42 − 32c62 )e7n + O(e8n )], f (yn ) = f ′ (α)[c2 e2n + (2c3 − 2c22 )e3n + (3c4 − 7c2 c3 + 5c32 )e4n

+ (24c3 c22 − 12c42 − 6c23 − 10c2 c4 + 4c5 )e5n + (34c4 c22

− 73c3 c32 + 28c52 + 37c2 c23 − 17c4 c3 − 13c2 c5 + 5c6 )e6n

+ (−160c23 c22 + 206c3 c42 − 104c4 c32 + 44c5 c22 − 64c62

+ 104c4 c2 c3 − 22c5 c3 + 6c7 − 16c2 c6 − 12c24 + 18c33 )e7n

+ O(e8n )],

(3.4)

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f ′ (yn ) = f ′ (α)[1 + 2c22 e2n + (4c2 c3 − 4c32 )e3n + (6c2 c4 − 11c3 c22 + 8c42 )e4n + (28c3 c32 − 20c4 c22 + 8c2 c5 − 16c52 )e5n

+ (−16c4 c2 c3 − 68c3 c42 + 12c33 + 60c4 c32 − 26c5 c22 + 10c2 c6 + 32c62 )e6n + (−84c2 c33 + 112c3 c4 c22

(3.5)

− 20c3 c2 c5 + 160c3 c52 + 36c4 c23 + 72c5 c32 + 12c2 c7

2

− 32c22 c6 − 24c2 c24 − 168c4 c42 − 64c72 )e7n + O(e8n )],

f ′′ (yn ) = f ′ (α)[2c2 + 6c2 c3 e2 + (12c23 − 12c3 c22 )e3n + (18c3 c4

− 42c2 c23 + 24c3 c32 + 12c4 c22 )e4n + (−12c4 c2 c3 − 48c4 c32

− 36c33 + 120c23 c22 + 24c5 c3 − 48c3 c42 )e5n + (72c2 c24

− 96c3 c4 c22 + 144c4 c42 − 54c4 c23 − 78c3 c2 c5 + 30c3 c6

+ 198c2 c33 − 312c23 c32 + 96c3 c52 + 20c5 c32 )e6n

(3.6)

+ (72c4 c2 c23 + 576c3 c4 c32 − 384c24 c22 + 96c4 c2 c5 − 384c4 c52 + 72c24 c3 − 132c5 c23 + 336c5 c3 c22 + 36c3 c7 − 96c3 c2 c6

+ 108c43 − 756c33 c22 + 768c23 c42 − 192c3 c62 − 120c5 c42 )e7n

+ O(e8n )], 3

f ′′′ (yn ) = f ′ (α)[6c3 + 24c2 c4 e2n + (48c4 c3 − 48c4 c22 )e3n + (72c24 − 168c4 c2 c3 + 96c4 c32 + 60c5 c22 )e4n

+ (240c3 c2 c5 − 240c5 c32 − 144c4 c23 + 480c4 c3 c22 − 240c2 c24 + 96c4 c5 − 192c4 c42 )e5n + (48c4 c2 c5 − 1320c3 c5 c22

+ 720c5 c42 + 240c5 c23 − 408c24 c3 + 672c24 c22 + 120c4 c6

+

+ 792c4 c2 c23 − 1248c4 c3 c32 + 384c4 c52 + 120c6 c32 )e6n

(3.7)

(−2400c5 c2 c23 + 5040c3 c5 c32 − 1056c4 c5 c22 + 480c2 c25 − 1920c5 c52 + 192c4 c5 c3 + 144c4 c7 − 384c4 c2 c6 − 288c34 + 2208c2 c24 c3 − 1728c24 c32 + 432c4 c33 − 3024c4 c23 c22 + 3072c4 c3 c42 − 768c4 c62 + 720c22 c6 c3 − 720c42 c6 )e7n + O(e8n )].

Using equations (3.1)-(3.7) in Algorithm 2.6, we get

xn+1 = α + (−2c3 c32 + 2c52 )e6n + (−12c23 c22 + 24c3 c42 − 12c62 )e7n + O(e8n ), which implies that en+1 = (−2c3 c32 + 2c52 )e6n + (−12c23 c22 + 24c3 c42 − 12c62 )e7n + O(e8n ),

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which shows that Algorithm 2.6 has at least order of convergence six.

4. Applications We present some examples to illustrate the efficiency of the Algorithm 2.6. We compare the Newton’s method (NM), the Halley’s method (HM), the Househ¨older’s method (HHM), the Abbasbandy’s method (AM), the Noor and Noor method (NNM) and the new iterative method (NIM) (Algorithm 2.6). We used ε = 10−15 . The following stopping criteria is used for computer programs: 1. |xn+1 − xn | < ε, 2. |f (xn+1 )| < ε, and consider f1 (x) = x3 + 4x2 − 10, f2 (x) = sin x2 − x2 + 1, x 1 f3 (x) = x2 + sin − , 5 4 x 2 f4 (x) = e − 4x , f5 (x) = (x − 1)3 − 1.

Table 1. Comparison of NM, HM, HHM, AM, NNM and NIM (f1 (x) = x3 + 4x2 − 10, x0 = 2.5) Method NM HM HHM AM NNM NIM

N 6 4 4 4 4 3

Nf 12 12 12 16 12 12

|f (xn+1 )| 1.335249e − 25 2.563378e − 36 6.046396e − 27 1.162312e − 28 2.563378e − 36 1.368292e − 82

xn+1 1.365230013414096845760806828980

Table 2-1. Comparison of NM, HM, HHM, AM, NNM and NIM (f2 (x) = sin x2 − x2 + 1, x0 = 1) Method NM HM HHM AM NNM NIM

N 6 4 5 4 4 3

Nf 12 12 15 16 12 12

|f (xn+1 )| 1.819126e − 25 2.527247e − 38 9.230984e − 28 6.138132e − 21 2.527247e − 38 1.960427e − 75

xn+1 1.404491648215341226035086817790

DYNAMICS OF MODIFIED ABBASBANDY’S METHOD...

Table 2-2. Comparison of NM, HM, HHM, AM, NNM and NIM (f2 (x) = sin x2 − x2 + 1, x0 = 3.6) Method NM HM HHM AM NNM NIM

N 6 4 4 5 4 3

Nf 12 12 12 20 12 12

|f (xn+1 )| 3.440228e − 21 8.722627e − 29 3.832086e − 23 5.041497e − 26 8.722627e − 29 8.669320e − 61

xn+1 1.404491648215341226035086817790

Table 3. Comparison of NM, HM, HHM, AM, NNM and NIM (f3 (x) = x2 + sin x5 − 14 , x0 = 2.1) Method NM HM HHM AM NNM NIM

N 7 4 5 5 4 3

Nf 14 12 15 20 12 12

|f (xn+1 )| 5.016083e − 27 2.433619e − 17 4.974916e − 37 5.316614e − 37 2.433619e − 17 2.195471e − 38

xn+1 0.409992017989137131621258376499

Table 4-1. Comparison of NM, HM, HHM, AM, NNM and NIM (f4 (x) = ex − 4x2 , x0 = 2) Method NM HM HHM AM NNM NIM

N 6 4 4 5 4 3

Nf 12 12 12 20 12 12

|f (xn+1 )| 9.667543e − 24 2.037198e − 20 9.307225e − 18 9.055775e − 40 2.037198e − 20 1.862615e − 67

xn+1 0.714805912362777806137622208112

Table 4-2. Comparison of NM, HM, HHM, AM, NNM and NIM (f4 (x) = ex − 4x2 , x0 = 0.5) Method NM HM HHM AM NNM NIM

N 5 3 4 4 3 2

Nf 10 9 12 16 9 8

|f (xn+1 )| 2.447553e − 21 2.356335e − 17 8.267416e − 35 9.567270e − 34 2.356335e − 17 1.194505e − 21

xn+1 0.714805912362777806137622208112

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Table 5. Comparison of NM, HM, HHM, AM, NNM and NIM (f5 (x) = (x − 1)3 − 1, x0 = 1.5) Method NM HM HHM AM NNM NIM

N 7 4 52 4 4 3

Nf 14 12 156 16 12 12

|f (xn+1 )| 1.091232e − 22 6.390950e − 24 7.662031e − 28 7.139947e − 32 6.390950e − 24 2.208268e − 35

xn+1 2.000000000000000000000000000000

Tables 1-5 shows the numerical comparisons of the Newton’s method (NM), the Halley’s method (HM), the Househ¨older’s method (HHM), the Abbasbandy’s method (AM), the Noor and Noor method (NNM) and the new iterative mathod (NIM) (Algorithm 2.6). The columns represent the number of iterations N and the number of functions or derivatives evaluations Nf required to meet the stopping criteria, and the magnitude |f (x)| of f (x) at the final estimate xn . 5. Polynomiographs Polynomials 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. 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 [12], who introduced the polynomiography. As a method which generates nice looking graphics, it was patented by Kalantari [13] in 2005. 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” [12]. An individual image is called a “polynomiograph”. Polynomiography combines both art and science aspects. Polynomiography gives a new way to solve the ancient problem by using new algorithms and computer technology. 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., the Newton’s method, the Halley’s method etc. The word “fractal”, which partially appeared in the definition of polynomiography. Both fractal images and polynomiographs can be obtained via different iterative schemes. Fractals are self-similar has typical

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structure and independent of scale. On the other hand, polynomiographs are quite different. The “polynomiographer” can be controlled the shape and designed in a more predictable way by using different iteration methods to the infinite variety of complex polynomials. Generally, fractals and polynomiographs belong to different classes of graphical objects. Polynomiography has diverse applications in math, science, education, art and design. According to Fundamental Theorem of Algebra, any complex polynomial with complex coefficients {an , an−1 , ..., a1 , a0 }: p(z) = an z n + an−1 z n−1 + ... + a1 z + a0

or by its zeros (roots) {r1 , r2 , ..., rn−1 , rn }:

p(z) = (z − r1 )(z − r2 )...(z − rn )

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 [12, 13, 18]. 6. Iteration During the last century, the different numerical techniques for solving nonlinear equation f (x) = 0 have been successfully applied. We define y n = xn −

f (yn ) f 2 (yn )f ′′ (yn ) f 3 (yn )f ′′′ (yn ) f (xn ) , x = y − − − . n+1 n f ′ (xn ) f ′ (yn ) 2f ′3 (yn ) 6f ′4 (yn )

This is Algorithm 2.6 for solving nonlinear equations. Let p(z) be the complex polynomial, then p(zn ) , n = 0, 1, 2, ..., yn = zn − ′ p (zn ) p(yn ) p2 (yn )p′′ (yn ) p3 (yn )p′′′ (yn ) zn+1 = yn − ′ − − , p (yn ) 2p′3 (yn ) 6p′4 (yn ) where zo ∈ C is a starting point, this is Algorithm 2.6 for solving nonlinear complex equations. The sequence {zn }∞ n=0 is called the orbit of the point zo converges to a root z ∗ of p then, we say that zo is attracted to z ∗ . A set of all ∗ such starting points for which {zn }∞ n=0 converges to root z is called the basin ∗ of attraction of z .

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6.1. 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. 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: |zn+1 − zn | < ε,

(6.1)

where zn+1 and zn are two successive points in the iteration process and ε > 0 is a given accuracy. In this paper we also use the stop criterion (6.1). 6.2. Applications The applications of the Algorithm 2.6 for solving nonlinear complex equations perturbs the shape of polynomial basins and makes the polynomiographs look more “fractal”. The aim of using the Algorithm 2.6 for solving nonlinear complex equations to create images that are quite new, different from images by the Newton’s method and Househ¨ older’s method free from second derivatives [23] and interesting from the aesthetic point of view. In this section we present some examples of polynomiographs for different complex polynomials equation p(z) = 0 and some special polynomials. 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. 6.2.1. Polynomiograph for z 2 − 1 = 0 Complex polynomial equation z 2 − 1 = 0, having two roots: 1, −1. The polynomiograph is presented in the following figure with two distinct basins of attraction to the two roots of the polynomial z 2 − 1 = 0. 6.2.2. Polynomiograph for z 3 − 1 = 0 Complex polynomial equation z 3 − 1 = 0, having three roots: 1, − 21 − √

√

3 2 I,

− 21 + 23 I. The polynomiograph is presented in the following figure with three distinct basins of attraction to the three roots of the polynomial z 3 − 1 = 0.

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Figure 1. Polynomiography for z 2 − 1 = 0.

Figure 2. Polynomiography for z 3 − 1 = 0.

6.2.3. Polynomiograph for z 4 − 1 = 0 Complex polynomial equation z 4 − 1 = 0, having four roots: -1, −I, I, 1... The polynomiograph is presented in the following figure with four distinct basins of attraction to the four roots of the polynomial z 4 − 1 = 0. 6.2.4. Polynomiograph for z 4 − z 3 + z 2 − z + 1 = 0 Complex polynomial equation z 4 − z 3 + z 2 − z + 1 = 0, having four roots: −0.309017−0.951057I, −0.309017+0.951057I, 0.809017−0.587785I, 0.809017+ 0.587785I. The polynomiograph is presented in the following figure with four distinct basins of attraction to the four roots of the polynomial z 4 − z 3 + z 2 − z + 1 = 0 = 0.

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Figure 3. Polynomiography for z 4 − 1 = 0.

Figure 4. Polynomiography for z 4 − z 3 + z 2 − z + 1 = 0.

6.2.5. Polynomiograph for z(z 2 + 1)(z 2 + 4) = 0 Complex polynomial equation z(z 2 + 1)(z 2 + 4) = 0, having five roots: 0, 0 − 1I, 0 + 1I, 0 − 2I, 0. + 2I. The polynomiograph is presented in the following figure with five distinct basins of attraction to the five roots of the polynomial z(z 2 + 1)(z 2 + 4) = 0. 6.2.6. Polynomiograph for z 5 − 1 = 0 √

5 Complex polynomial equation z 5 − 1 = 0, having five roots: 1, −1 4√ + 4 √+ √ √ √ √ √ √ √ √ √ √ √ √ √ I 2 5− 5 −1 I 2 5+ 5 −1 5 , 4 − 45 + I 2 45− 5 , −1 , 4 + 45 − I 2 45+ 5 . 4 4 − 4 − 4 The polynomiograph is presented in the following figure with five distinct basins of attraction to the five roots of the polynomial z 5 − 1 = 0.

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Figure 5. Polynomiography for z(z 2 + 1)(z 2 + 4) = 0.

Figure 6. Polynomiography for z 5 − 1 = 0.

6.2.7. Polynomiograph for z 20 − 1 = 0 Complex polynomial equation z 20 − 1 = 0. The polynomiograph is presented in the following figure with twenty distinct basins of attraction to the twenty roots of the polynomial z 20 − 1 = 0. References [1] S. Abbasbandy, Improving Newton-Raphson method for nonlinear equa- tions by modified Adomian decomposition method, Appl. Math. Comput., 145 (2003), 887-893, doi: 10.1016/S0096-3003(03)00282-0. [2] A. Ali, M.S. Ahmad, W. Nazeer, M. Tanveer, New modified two-step jungck iterative method for solving nonlinear functional equations, Sci. Int. (Lahore), 27 (2015), 29592963.

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