Optimal Algorithms for Nonlinear Equations with Applications and Their Dynamics

In the present work, we introduce two novel root-finding algorithms for nonlinear scalar equations. Among these algorithms, the second one is optimal according to Kung-Traub’s conjecture. It is established that the newly proposed algorithms bear the fourthand sixth-order of convergence. To show the effectiveness of the suggested methods, we provide several real-life problems associated with engineering sciences. *ese problems have been solved through the suggested methods, and their numerical results proved the superiority of these methods over the other ones. Finally, we study the dynamics of the proposed methods using polynomiographs created with the help of a computer program using six cubic-degree polynomials and then give a detailed graphical comparison with similar existing methods which shows the supremacy of the presented iteration schemes with respect to convergence speed and other dynamical aspects.


Introduction
A huge number of complicated problems in mathematics and engineering disciplines are directly connected to the solution of transcendental and algebraic nonlinear equations of the form: where the real-valued function φ is defined on the open connected set. Mostly, the direct solution of these problems is not possible to find via analytical methods, and ultimately, we have to move towards the iterative algorithms. By an algorithm, we mean a sequence of finite number of steps to achieve the required goal (solution) of the given problem. e repetition of these steps is called iterations. In the process of iterative algorithms, we always need a starting point to initialize the iteration process. is starting point is usually called the initial guess that is rectified after each iteration till the required accuracy is gained.
In literature, there is a plethora of root-finding iterative algorithms for the problems related to the nonlinear equations. e oldest, classical, and most widely used algorithm was suggested by Newton-Raphson in 1690 [1]. Geometrically, the derivation of Newton's algorithm was purely based on the concept of slope of the line. is method requires two evaluations per iteration and possesses the quadratic convergence order. For many years, Newton's method has been implemented successfully for root-finding of nonlinear problems. After that, Gutierrez and Hernández [2] presented a new family of Chebyshev-Halley type methods in Banach spaces which were utilized for root-finding of nonlinear problems. In 1993, Argyros et al. [3] discussed the applications of Halley method in Banach space. After few years, Chun [4] constructed Newton-like iteration methods which were purely designed for finding one-dimensional nonlinear equations' solution. After the construction of one step Newton-like iterative algorithms, a huge class of researchers tried to modify it and suggested a broad range of root-finding algorithms with the help of different mathematical techniques and established a class of multistep algorithm. For further details of the multistep algorithm, one can see Amiri et al. [5], Behl et al. [6], Naseem et al. [7], and Ozyapici [8]. e motivation behind these modifications is to attain higherorder and more efficient algorithms.
Ostrowski [9] and Traub [10] in the twentieth century introduced the concept of multistep iteration schemes and proposed two-step fourth-order iteration schemes with Newton's iteration method as a predictor. In 2006, Aslam Noor and Inayat Noor [11] introduced some new three-step iterative schemes and proved their third-order convergence. After that, Golbabi and Javidi [12] in 2007 presented a new cubically convergent method whose derivation is totally based on the homotopy-perturbation method. By utilizing the new series expansion of the nonlinear function, Noor et al. [13], in 2012, constructed and then analyzed some novel two-step iteration methods and discussed some special cases. ese suggested schemes possessed the convergence of quadratic and cubic orders and were actually the modified form of Newton's algorithm. Kumar et al. [14] presented a novel class of sixth-order parameterbased methods for finding zeros of one-dimensional nonlinear equations in 2018. In 2019, Said Solaiman et al. [15] proposed derivative free optimal fourth-order and eighth-order versions of King's approach by combining the composition technique with rational interpolation, as well as the Pade's concept of approximation. In 2020, Chand et al. [16] developed some novel PotraPtak type optimal sixth-and eighth-order iteration methods by utilizing the idea of weight functions on the PotraPtak method for determining the approximate zeros of nonlinear models and applied them on some real-life engineering problems. e authors also analyzed the stability of the suggested schemes via basins of attraction for some cubic and quadratic polynomials. Naseem et al. [17] recently constructed and analyzed some novel ninth-order iteration schemes by implementing the idea of variational iteration and then investigated the dynamical behaviour via polynomiographs of different complex polynomials. e main contributions of the present research are given as follows: (i) We developed and examined two novel predictor corrector type iterative techniques, namely, Algorithm 1 and Algorithm 2, in which Newton's method is used in the predictor step. (ii) We demonstrated that these newly created approaches have fourth-and sixth-order of convergence and are more efficient than the other wellknown iterative methods of the same type. (iii) e suggested approaches were used to solve several test cases in order to evaluate their validity and accuracy. (iv) We compare the polynomiographs of newly proposed techniques to those of existing methods of the same category in terms of time and other dynamical aspects. e exhibited polynomiographs contain highly fascinating and attractive patterns that represent many polynomial characteristics. e remaining sections of the paper are arranged as follows: In Section 2, we have constructed two novel algorithms. e convergence criteria for the constructed algorithms have been discussed in Section 3. In Section 4, we solved different test problems for assuring the validity of the constructed methods. Section 5 includes the graphical properties of the constructed algorithms, and Section 6 contains the conclusion.

Construction of New Optimal Root-Finding Algorithms
By employing the technique of modified homotopy perturbation, Golbabai and Javidi [12] introduced the following iteration formula: which is a third-order iteration method, usually known as Javidi's method. In [18], Rafiq and Rafiullah considered Newton's iteration method in the predictor step and presented a new two-step Javidi's iteration method given as To make it optimal, we consider the following approximations of first and second derivatives: Using the above approximations in (3), we gain new optimal root-finding algorithms having the following iterative form: Algorithm 1.For a given x 0 , compute the approximate solution x p+1 by the following iterative schemes: which is fourth-order optimal root-finding algorithm for nonlinear scalar equations and it utilizes three functional evaluations per iteration. It should be noted that all the terms that appeared in the denominator of Algorithm 1 must not 2 Complexity be zero; otherwise, the method will fail to find the approximate solution to the given problem. To achieve better convergence, we utilize the same idea as described above and add one more step as Newton's method which results in the following iterative method: e above iteration scheme gains optimal order but does not fulfill the Kung and Traub's conjecture [19]; to fulfill this conjecture, we consider the following approximations: where G � G u p , v p , w p , With the help of the above approximations in (6), we are able to suggest the following algorithm: Algorithm 2.For a given x 0 , compute the approximate solution x p+1 by the following iterative schemes: which is a three-step optimal root-finding algorithm, having sixth convergence order and utilizing only four functional evaluations per iteration. One must keep in mind that all the terms that appeared in the denominator of Algorithm 2 must not be vanished at the initial guess in the given domain; otherwise, the method will not work properly to find the approximate solution of the given problem.

Convergence Analysis
is section of the paper contains the convergence analysis of the suggested iteration methods.

Theorem 1. Suppose α be the actual root of equation
Proof. To prove the theorem, suppose α be the exact root of the equation φ(x) � 0 and e p be p th-iteration's error, where e p � x p − α, and with the help of Taylor's series expansion around x � α, we obtain φ ′ x p � φ ′ (α) 1 + 2d 2 e p + 3d 3 e 2 p + 4d 4 e 3 p + 5d 5 e 4 p + 6d 6 e 5 p + 7d 7 e 6 p + O e 7 p , where With the help of (10) and (11), we get Complexity With the help of (10), (13), and (14), we get Using equations (10)- (15) in Algorithm 1, we obtain which implies that e above equations confirms that Algorithm 1 is of fourth-order convergence.
Proof. From equations (10)- (15) with the same assumptions of the previous theorem, we have 4 Complexity Using equations (10)- (24) in Algorithm 2, we get which implies that e above equations confirm that Algorithm 2 is of sixth-order convergence.

Numerical Comparisons and Applications
To demonstrate the applicability and effectiveness of our newly devised iterative approaches, we present five realworld engineering problems and one very important and well-known nonlinear problem in this section. e devised iterative approaches are compared to the following existing two-step iterative algorithms:

Noor's Method One (NM1).
For the given initial guess x 0 , calculate the approximate solution x p+1 using the iteration schemes as follows: which is the second-order convergent method, known as Noor's method one [11] for solving nonlinear scalar equations.

Chun's Method (CM).
For a given initial guess x 0 , determine the approximate root x p+1 with the iteration schemes: which is the fourth-order convergent Chun's method [20] for solving nonlinear scalar equations.

Chand's Method (CHM).
For the given initial guess x 0 , calculate the approximate solution x p+1 using the iteration schemes: which is the fourth-order two-step Chand's method [16] for solving nonlinear scalar equations.

Noor's Method Two (NM2).
For the given initial guess x 0 , calculate the approximate solution x p+1 using the iteration schemes: 6 Complexity which is three-step cubic-order Noor's method two [13] for solving nonlinear scalar equations.

Yun's Method (YM).
For the given initial guess x 0 , calculate the approximate solution x p+1 using the iteration schemes: which is the three-step Yun's method [21] for solving nonlinear scalar equations, having the convergence of fourth order. We evaluate the following test examples to conduct a numerical comparison of the above-described methods with our proposed algorithms: Example 1. Kinetic problem equation. e equation of kinetic problem has the following form: Where Trepresents the temperature of the given system. (32) has been derived from the stirred reactor with the cooling coils [22]. By taking T � x, (32) may be rewritten in form of the following nonlinear function: which can be used to find the temperature of the system. We start the iteration process with the initial guess x 0 � 430, and the related results from various iteration techniques are shown in Table 1.
Example 2. Planck's radiation law. e Planck's radiation law [23] is used to compute the energy density within an isothermal black body with the standard form as Assume that we want to determine the wavelength sigma 1 that corresponds to maximal energy density φ(σ 1 ).
We use x � cP/σkT to turn the aforementioned problem into a nonlinear equation, which has the following nonlinear equation: which can be converted into the form of nonlinear function as follows: e maximal wavelength of the radiation is represented by the estimated root of the aforementioned function φ 2 . To begin the iteration process, we take the starting guess x 0 � 0.2, and the relevant results from various iteration methods are shown in Table 2. e equation of adiabatic flame temperature has the following form: where we take ΔH � −57798, a 1 � 7.256, a 2 � 0.002298, and a 3 � 0.00000283. For more information, one may see [24,25] and the references are cited therein. e aforementioned function φ 3 is actually a third-degree polynomial, and according to algebra's fundamental theorem, it must have three unique roots (zeros) and α � 4305.30991366612556304019892945 is a simple one among them that we estimated using the proposed algorithms with the starting guess x 0 � 2000, and the numerical results are presented in Table 3.

Example 4. Beam designing model.
We consider the problem of beam positioning from [26], which yields a nonlinear function as e given function φ 4 is actually a four-degree polynomial, and it must have precisely four roots (zeros) in the light of fundamental theorem of algebra. We choose the starting guess x 0 � −0.75 to approximate the required root using the proposed algorithms, and the numerical results are shown in Table 4.

Example 5. Open channel flow problem.
In fluid dynamics, Manning's equation (27) deals with the water flow with the following standard form: In (39), the symbol s stands for the slope, a stands for the area, r stands for the hydraulic radius, and n stands for Manning's roughness coefficient. For a channel with the rectangular-shape of width w and depth x, we may have Using these values in (39), we obtain In order to compute water's depth in a channel, we rewrite (41) in the following nonlinear form: In (42), the parameters haven been chosen as w � 4.572m, s � 0.017, F � 14.15m 3 /s, and N � 0.0015. We start the iteration process with the initial guess x 0 � 0.1, and the related results from various iteration techniques are shown in Table 5.

Example 6. A well-known nonlinear problem.
To analyze the proposed methods, we consider the following very important and well-known nonlinear problem: x � tan x.
(43) e above equation in the form of nonlinear function φ 6 can be rewritten as  Method  Method  Method We start the iteration process with the initial guess x 0 � 1.0, and the related results from various iteration techniques are shown in Table 6. e machine used for computing numerical results has the following specifications: (i) 64 bit operating system (ii) x64-based processor with Core(TM) m3-7Y30 CPU@1.00 GHz 1.61 GHz (iii) 8 GB of memory We use the accuracy ε � 10 − 15 in the stopping criteria |x p+1 − x p | < ε for the all aforementioned problems. We used the computer application Maple 13 to compute all of the numerical results and can be observed in Tables 1-6. Tables 1-6 represent the detailed comparison of the suggested root-finding methods with the other above-described methods for the engineering and arbitrary problems φ 1 -φ 6 . In the columns of the above-presented tables, N stands for the consumption of the iterations for different methods, |φ(x)| stands for the modulus value of φ(x), x p+1 represents the final estimation, |x p+1 − x p | stands for the positive distance between the two consecutive estimations, and the last columns gives us the information about the CPU time consumption in seconds for different methods in comparison.
e careful examination of the obtained results in Tables 1-6 certifies that the proposed root-finding algorithms are showing better efficiency and performance which justified the supremacy of the suggested root-finding algorithms with respect to CPU-time consumption, accuracy, convergence-speed, no. of iterations, and computational order of convergence against the other comparable methods.

Dynamical Representation
In this section, we investigate the dynamical aspects of different algorithms with the aid of polynomiographs created through different algorithms for complex polynomials of different degrees. To generate polynomiographs of different complex polynomials by means of a computer program, we have to choose an initial rectangle R which contains the polynomial's roots.
en, corresponding to each starting point w 0 in the region, we execute an iterative process and then colour the point corresponding to w 0 that relies on the approximate convergence of the truncated orbit to a root. e discretization of R is completely responsible for the image's resolution quality. For instance, if we discretize R into a 2000 by 2000 grid, the output is a highresolution picture.
We know that any complex polynomial q having degree n has exactly n-roots, and from the fundamental theorem of algebra, it may be uniquely defined as q(w) � c n w n + c n−1 w n− 1 + · · · + c 1 w + c 0 .
or by its zeros (roots) w 1 , w 2 , . . . , w n−1 , w p : where c n , c n−1 , . . . , c 1 , c 0 are the complex coefficients. e iterative algorithms can be easily applied to the both representations of the complex polynomial q. e polynomial's degree depicts the number of basins of attraction. e location of basins can be managed by changing the position of roots in the complex plane manually. e polynomiographs' colours depends upon the no. of iterations needed to attain the approximate solution of some polynomial with a given accuracy and a chosen scheme of iteration. e main algorithm for drawing polynomiographs is given in Algorithm.
In Algorithm 3, the convergence test (w p + 1, w p , ϵ) would be considered true in case of convergence and vice versa. e following is the standard form of the widely used convergence test: In (47), w p and w p+1 are the consecutive estimations in the iteration procedure, and the symbol ε > 0 stands for the accuracy. In this article, we also use the stopping criteria (47). We created visually appealing and intriguing polynomiographs using newly developed root-finding methods and compared them with the polynomiographs of the other similar methods. e colour of polynomiographs is determined by the no. of iterations required to estimate the roots of a polynomial with a certain precision ε. A huge number of similar graphics may be made by changing the value of the parameter k, where k specifies the maximum no. of iterations. e work on polynomiography was first initiated by Kalantri [28][29][30] who described its artistic applications in different fields of science and arts. Gdawiec et al. [31,32] put forward the work of Kalantri and presented fractal patterns of polynomial root-finding methods. Scott et al. [33], in 2011, worked on the basis of attraction for different existing methods and compared them with respect to their basis. In Table 5: Numerical comparison among different algorithms for the problem φ 5 .
Method this sense, the polynomiography gives us a new way to analyze the graphical behaviors of different existing methods in the literature. We investigate the following complex polynomials for the aim of generating polynomiographs using the suggested methods, and we compare them with other well-known twostep iteration methods.
e colormap that has been used for the coloring of iterations in the generation of polynomiographs is presented in Figure 1: Polynomiographs corresponding to polynomial q 1 (w) through various numerical algorithms.
In this example, we investigate and compare the dynamical results obtained through different iteration schemes with our presented algorithms by considering the cubic polynomial w 3 − 1 which possesses three distinct simple zeros: 1, −1/2 − � 3 √ /2i, and −1/2 + � 3 √ /2i. We executed all the algorithms to achieve the simple zeros of the considered polynomials, and the results can be visualized in Figures 2-8.
Example 8. Polynomiographs corresponding to polynomial q 2 (w) through various numerical algorithms.
is example includes the dynamical comparison of the suggested iteration schemes with different similar-nature iterative algorithms by taking the cubic complex polynomial Example 9. Polynomiographs corresponding to polynomial q 3 (w) through various numerical algorithms.
We consider the complex polynomial w 3 + w.i + 1 in this experiment for analyzing the behaviors of different iteration schemes graphically. For this purpose, we generated the polynomiographs of the considered polynomials whose simple zeros are (1/6)3 [ Example 10. Polynomiographs corresponding to polynomial q 4 (w) through various numerical algorithms.
In this example, we show the dynamics of different iteration schemes in the form of polynomiographs by taking the complex polynomial w 3 − w 2 + w − 1, whose simple zeros are 1, i, and −i that can be visualized easily on the complex planes of the corresponding polynomiographs that are shown in Figures 23-29.
Example 11. Polynomiographs corresponding to polynomial q 5 (w) through various numerical algorithms.
In the eleventh example, we take the polynomial w 3 + w 2 + w + 1, having simple zeros −1, i, and −i. To draw the polynomiographs, we executed all the iteration schemes with the help of computer program, and the corresponding graphical objects can be visualized in Figures 30-36.
Example 12. Polynomiographs corresponding to polynomial q 6 (w) through various numerical algorithms.
In the last and final experiment, we show the graphical representations of different iteration schemes by generating Colour w 0 by means of colormap.               In all six experiments, we considered the different cubic complex polynomials for examining the graphical aspects of the suggested iteration schemes. In all the obtained images, the regions of convergence for the suggested iteration schemes possess the larger convergence areas than the other      comparable methods which certified the efficiency and better convergence of our proposed algorithms. e colour tones represent the performance and efficiency of the considered algorithm used to build the polynomiograph. e two important aspects which can be exhibited by these graphical objects are the convergence-speed and the dynamics of the      14 Complexity considered iteration techniques used to build these graphs. e first may be shown by examining the image's colour tones. e darkness of the colours in the presented images demonstrates fast convergence with less number of iterations, i.e., the darker the image, the more efficient the approach, and the given images demonstrate the superiority of

Conclusions
In this study, we have established and analyzed two novel optimal root-finding algorithms for nonlinear functions. e convergence criterion of the presented algorithms has been discussed and verified that the suggested iterative algorithms bear fourth-and sixth-order convergence. To exhibit the applicability of the suggested algorithms, some real-life engineering applications have also been added and solved whose numerical results confirmed that the proposed algorithms are time-efficient and consumed less iterations to achieve the required solution and more accurate to the exact solution. e dynamics of the provided methods demonstrating the greater convergence area in comparison with the other comparable methods. ese dynamics also highlighted the proposed algorithms' quicker convergence speed and other dynamical properties, proving their superiority over the others in comparison.

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

Conflicts of Interest
e authors do not have any conflicts of interest.