A New Root-Finding Algorithm for Solving Real-World Problems and Its Complex Dynamics via Computer Technology

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Introduction
e role of computers in the fields of applied mathematics cannot be denied in the modern age. Using different computer programs such as Mathematica, Matlab, and Maple, a plethora of different types of complex problems can be solved easily. In recent years, the mathematicians employed the excessive use of computers in different fields of mathematics.
e most important among them is the polynomial's root finding which has played a significant role in applied and computational mathematics and covers many other fields of modern sciences. ere exist several problems in engineering which arise in the form of nonlinear equations. For solving such type of engineering problems, we need iterative algorithms because in most of the cases, analytical methods do not work. e well-known classical iterative algorithm was given by Newton [1] in the last of fifteenth century. In the modern age, the researchers improved the order and efficiency of the existing methods and introduced multistep algorithms. For more details, see [2][3][4][5][6][7][8][9][10][11] and the references cited therein. Usually, multistep algorithms possess greater convergence with higher computational cost due to the involvement of higher derivatives which is the downside of these algorithms. It is really a tough task to balance the computational cost and the convergence order because if we increase one of them, the other would be decreased.
In the last few years, the researchers modified the existing iteration schemes by employing different mathematical techniques and suggested a new variety of multistep iterative methods. In [12], the author introduced the predictor-corrector method given as v p � u p − f u p f ′ u p , p � 0, 1, 2, 3, . . . , which is quartic-order algorithm suggested by Traub (TM) for root finding of scalar nonlinear equations. Noor et al. [13], in 2012, introduced an iterative algorithm of the form: which is known as Noor's method (NRM) for root finding of scalar nonlinear equations. In 2010, Zhanlav et al. [14] introduced a new three-step iterative algorithm given as which is quartic-order algorithm for root finding of scalar nonlinear equations known as Zhanlav's method (ZM). In the present research article, we introduced a new fourth order and derivative-free algorithm for solving engineering and arbitrary problems in the form of scalar nonlinear functions. e construction of this algorithm is based upon the forward-and finite-difference schemes on Golbabai-Javidi's method. We also certified that the designed algorithm has fourth-order convergence and then applied to different real-world engineering and arbitrary problems to demonstrate its applicability among the other fourth-order algorithms in the literature. e dynamical comparison of the designed algorithm with the other comparable ones has been also presented via computer technology using the program Mathematica 12.0. e rest of the paper is divided as follows. An efficient and derivative-free algorithm has been constructed in Section 2. e convergence criterion of the designed algorithm has been discussed in Section 3. In Section 4, nine arbitrary and engineering problems have been solved. e graphical analysis of the designed algorithm has been presented in Section 5. Finally, the conclusion of the paper is given in Section 6.

Main Results
Let f: D ⟶ R, D ⊂ R be a function in one variable; then, by employing the modified homotopy perturbation method, Golbabai and Javidi [15] introduced the following rootfinding algorithm: which is known as Golbabai-Javidi's method of cubic convergence [15]. In [16], the authors modified it and suggested a two-step iteration scheme of the following form: which is a two-step iteration scheme for calculating zeros of nonlinear scalar equations. e major drawback of the above algorithm is its high computational cost per iteration as it requires first and second derivatives for its execution. To lower its computational cost and make it more effective, we approximate its derivatives and make it derivative-free so that it can be easily applied on those nonlinear functions whose first derivative becomes infinite or does not exist. To approximate f ″ (v), we employ the finite-difference approximation as Now, we apply the forward-difference scheme to approximate first derivatives as follows: Using (7) and (8) in (9), Using (7)-(9) in (5), we are able to write Algorithm 1.

Complexity
Algorithm 1. For a given initial guess u 0 , determine the approximate solution u p+1 by the following iteration schemes: which is a novel derivative-free two-step iterative algorithm for calculating approximate zeros of nonlinear functions in one variable. e basic and important feature of the presented algorithm is its applicability area because it also covers such nonlinear functions in which first and second derivatives do not exist. e replacements of the first and second derivatives reduce its computational cost per iteration which yield higher efficiency index against the other similar iteration schemes. e obtained numerical results of test examples show better performance of the proposed algorithm against the other comparable ones in literature.

Convergence Analysis
In the present section, we proved the convergence order of the proposed method, i.e., Algorithm 1.
is sufficiently smooth near the exact root β; then, Algorithm 1 possesses quartic-order convergence. Moreover, it satisfies the following error equation: Proof. To prove the theorem, suppose e p as the error at pth iteration; then, e p � u p − β and the Taylor series about u � β implies where Using (13) and (14), we obtain Complexity 3 Using equations (13)- (18) in Algorithm 1, we get the following equality: which implies that e above equation established that Algorithm 1 is of quartic-order convergence.

Numerical Comparisons and Applications
In this section, we include five arbitrary and four engineering problems to demonstrate the validity, accuracy, and robust performance of the newly designed iteration scheme. We compare the suggested algorithm with Noor's method [13], Traub's method [12], and Zhanlav's method (ZM) [14] by considering some nonlinear problems.

Example 1. Blood rheology model.
Blood rheology is a branch of science that works to study the physical and flow properties of the blood [17]. Blood is actually a non-Newtonian fluid and treated as Caisson Fluid.
e model of Caisson fluid shows that the flow of simple fluids in a tube is such a way that the center core of the fluids will move as a plug with little deformation and velocity gradient occurs near the wall. To study the plug flow of Caisson fluids flow, we consider the following function in the form of nonlinear equation as where flow-rate reduction is computed by H. Using H � 0.40 in (16), we have To solve f 1 , the initial guess has been chosen as u 0 � 0.9 for starting the iteration process, and the results are given in Table 1.

Example 2. Finding volume from van der Waal's equation.
In engineering, the famous and well-known van der Waal's equation is used to examine gases behaviors [18] which was introduced by van der Waal: By assuming feasible values of the appearing parameters in (23), we obtain the following nonlinear problem: where u denotes the volume and may be found easily by solving f 2 . Since f 2 is a polynomial with cubic degree, so clearly three roots exist and the only one among them is feasible positive real root 1.9298462428 as the volume of the gas is always positive. To start the iteration process, we take u 0 � 0.10, and the results are inserted in Table 2.
Example 3. Open channel flow problem. e flowing water in the open channel with the condition of uniform flow is described in the well-known Manning's equation [19]: e parameters r, s, and a in (25) denote hydraulic radius, slope, and area of the channel, respectively, and N is Manning's roughness coefficient. For a rectangular-shaped channel with depth u and width w, we have Using these values in (25), we obtain For determining the depth of water, the above relation may be written as We assign the values to the parameters as G � 14.15 m 3 /s, w � 4.572m, s � 0.017, and N � 0.0015. We select the starting point u 0 � 4.50 to initialize the iteration process, and the results are inserted in Table 3.

Example 4. Plank's radiation law.
For finding the energy density in the isothermal black body, Planck's radiation law is used which was introduced by Planck [20] in 1914 having the following mathematical expression: To determine wavelength σ 1 against the peak value of energy density f(σ 1 ), we convert equation (29) in nonlinear equation by considering u � cP/σkT, given as below: which can be converted into the form of nonlinear function as follows: e approximated root of f 4 denotes the highest wavelength of the radiation. We select u 0 � 2.10 as the initial guess in the iteration process, and the results are inserted in Table 4.

Example 5. Arbitrary problems.
To examine the numeric behavior of the designed algorithm, we take five arbitrary problems, and the results are inserted in Table 5.
In Tables 1-5, a detailed comparison of well-known iterative algorithms with the suggested algorithm has been presented. e columns of the tables give us details about the quantity of consumed iterations, the final approximated root, the absolute value of the function at that root, and the positive difference between the two consecutive approximations.
e obtained results of the test examples which are summarized in the form of Tables 1-5 show the performance of the proposed algorithm, and from these results, and we can claim the robust performance of the designed algorithm in terms of accuracy, speed, number of iterations, computational cost and can say that it is superior to the other comparable algorithms.

Complex Dynamics
In this section, we study the complex dynamics of the suggested root-finding algorithm for different complex polynomials in the form of polynomiographs which are generated in the process of polynomiography. e term polynomiography was first introduced in 2005 by Bahman Kalantari [21,22] who defined it as a process of drawing aesthetically pleasing graphical objects by employing the mathematical convergence properties of iteration functions. By an iteration function, we mean such a function that is composed onto itself again and again such as Newton's algorithm [23], Halley's algorithm [24], and Householder's algorithm [25]. As a result of polynomiography, the generated graphical objects are known as polynomiographs. To plot the polynomiographs on the complex plane C via computer program by considering different complex polynomials, we take a rectangle R ∈ C of size [− 2, 2] × [− 2, 2], with the accuracy ε � 0.001 and maximum number of iterations L � 20.
is rectangle contains the roots of the considered polynomial, and corresponding to the starting point z 0 in R, we initialize the iteration process and assign a color to the point corresponding to z 0 . e black color is assigned to those particular points at which the algorithm fails to converge. e quality and resolution of the generated graphical objects are depended on the discretization of R, i.e., if we discretize R into the grid of 2000 × 2000, the plotted polynomiographs will have better resolution and image quality.
We know that if q is an nth-degree polynomial, then it must possess nth number of zeros according to the theorem of algebra and may be expressed as q(z) � d n z n + d n− 1 z n− 1 + · · · + d 1 z + d 0 .
If r 1 , r 2 , . . . , r n− 1 , r n are the roots (zeros) of q, then (33) may be rewritten as where d n , d n− 1 , . . . , d 1 , d 0 are the complex coefficients. Any algorithm that involves iteration process may be applied to both above-described expressions of q for plotting graphical objects. e general algorithm for plotting polynomiographs is given in Algorithm 2.
We consider an algorithm to be converged if the convergence test (z p+1 , z p , ϵ) returns TRUE and diverged if it returns False in Algorithm 1. e standard test for studying the convergence or divergence of an algorithm is given as where ε > 0 is the accuracy and z p and z p+1 are the two consecutive estimations in the iteration process. In this   Complexity study, we also consider equation (20) as a stopping criterion. A variety of different color graphical objects can be plotted by changing the parameters ε and k and the iteration scheme.
Example 6. Polynomiographs for q 1 through various iteration schemes.
In the first example, we investigate and compare the dynamical results obtained through different iteration schemes with the newly designed quartic-order algorithm for the quadratic-degree polynomial q 2 − 1 which possesses two distinct zeros: 1 and − 1. We executed all comparable algorithms to achieve the simple roots, and the results can be visualized in Figure 2.
Example 7. Polynomiographs for q 2 through various iteration schemes.      Input: q ∈ C[Z] is a complex polynomial, A ⊂ C is area in which polynomiographs are drawn, L is upper bound of iterations, ϕ is iterative scheme, ϵ is accuracy for the stopping criterion, colormap [0, . . . , C − 1] is colormap with C colors. Output: polynomiograph in the area in A for the complex polynomial q.
ALGORITHM 2: Generation of the polynomiograph.

Complexity
In the second example, we investigate and compare the dynamical results obtained through different iteration schemes with the newly designed quartic-order algorithm for the cubic-degree polynomial q 3 − 1 which possesses three distinct zeros: 1, − 1/2 − � 3 √ /2i, and − 1/2 + � 3 √ /2i. We executed all comparable algorithms to achieve the simple roots, and the results can be visualized in Figure 3.
Example 8. Polynomiographs for q 3 through various iteration schemes.
In the third example, we consider quartic-degree polynomial q 3 , having roots 1, − 1, i, and − i. We drew the aesthetically pleasing graphical objects by executing all comparable algorithms, and the results can be visualized in Figure 4.
In Examples 6-8, we ran all the comparable algorithms for drawing the aesthetically pleasing polynomiographs with the help of a computer program Mathematica 12.0. From the obtained graphical objects, we can easily examine the graphical behavior and stability of different algorithms. It should be noted that the convergence region of the newly designed algorithm is much larger as compared to other ones. e colors' shades exhibit the performance of that algorithm by which the polynomiograph is drawn. Two main characteristics can be revealed through these graphical objects which are the convergence speed and the dynamics of the considered iteration schemes by which these graphs have been generated. e first one can be depicted by analyzing the shades of colors of the image. e richness of the color in the graphical objects shows robust convergence with less iterations. e second property can be investigated by visualizing the variation of the colors of the drawn polynomiographs. e small variation's regions indicate the low dynamic zones and the dynamics are high for the zones of large variation of colors. e black color in the presented graphical objects showing the divergence zone of the algorithm, i.e., the zone where the solution is not possible to obtain for given accuracy and number of iterations.
e identical color zones represent the same number of iterations utilized by different iteration schemes for finding the solution up to given accuracy, and their graphical view to the contour lines on the map is similar.

Concluding Remarks
In the present study, a new efficient and derivative-free algorithm for computing roots of nonlinear equations has been developed by employing the forward-and finite-difference schemes on Golbabai-Javidi's. e convergence analysis of the suggested algorithm has been discussed and established that the suggested algorithm possesses the quartic-order convergence. To demonstrate the robust performance and validity of the presented algorithm, we assumed some arbitrary and engineering problems, converted them in the form of nonlinear functions, and then solved. e numerical results, contained in Tables 1-5, showed that the newly designed algorithm is superior to the other comparable algorithms with respect to different numerical terms such as convergence speed, accuracy, and computational order of convergence. e robust working of the new algorithm can also be judged by observing the accuracy of successive approximations which is much better against the other comparable ones. To study the complex dynamic nature of the suggested algorithm, polynomiographs have been drawn for some complex polynomials by utilizing the computer program Mathematica 12.0. e obtained graphical objects are novel and aesthetically pleasing, revealing the better convergence speed and other graphical characteristics of the designed algorithm over the other comparable ones. Using the same idea of this study, one can obtain a new family of derivative-free algorithms for computing roots of nonlinear equations.

Data Availability
All data required for this paper are included within the article.