# Some Real-Life Applications of a Newly Designed Algorithm for Nonlinear Equations and Its Dynamics via Computer Tools

Authors: Amir Naseem; M. A. Rehman; Jihad Younis
Journal: Complexity
 (2021)

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## Abstract

In this article, we design a novel fourth-order and derivative free root-finding algorithm. We construct this algorithm by applying the finite difference scheme on the well-known Ostrowski’s method. The convergence analysis shows that the newly designed algorithm possesses fourth-order convergence. To demonstrate the applicability of the designed algorithm, we consider five real-life engineering problems in the form of nonlinear scalar functions and then solve them via computer tools. The numerical results show that the new algorithm outperforms the other fourth-order comparable algorithms in the literature in terms of performance, applicability, and efficiency. Finally, we present the dynamics of the designed algorithm via computer tools by examining certain complex polynomials that depict the convergence and other graphical features of the designed algorithm.


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## Body

## 1. Introduction

The 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, mathematicians employed the excessive use of computers in different branches of Mathematics especially in the determination of approximated roots of the transcendental and nonlinear algebraic equations which have played an important role in different branches of computational and applied mathematics. In many engineering disciplines, a lot of problems exist which can be easily converted into nonlinear forms by employing different mathematical techniques. Analytical methods cannot find the solution needed for these problems, and therefore, we need iterative algorithms for solving out these problems. To execute an iterative algorithm, we always need a starting point (initial guess) which is refined after every iteration, and we find the approximated root up to the required accuracy after some finite iterations. The convergence rate and convergence order of an iterative algorithm are relied upon the selection of that starting point. Some of the most popular and ancient iterative algorithms are given in [1–8] and the references are cited therein. In the 15th century, Newton [1, 2] introduced a quadratic-order root-finding algorithm which has been used successfully for many years. Over the time, many experts worked on iterative algorithms and brought several modified versions of Newton’s algorithm with higher-order convergence which involve predictor and corrector steps and are often referred to as multistep iterative algorithms. For more information, one can see [9–20] and the references are cited therein. In general, the convergence order of a multistep algorithm is higher because of predictor and corrector steps, but it results in a higher computational cost which is the downside of these algorithms. It is really difficult to handle the cost of computing and the convergence rate of an algorithm because these two terms are inversely proportional to each other.Over the past few years, mathematicians have focused on the aforementioned issues and have tried to design some new iterative algorithms with higher convergence and low cost of computing by employing several mathematical methods. In [21], the authors introduced a new two-step Halley’s method with sixth order convergence and then replaced its second derivative for reducing computing cost and proposed a new fifth order second derivative free algorithm. In [22], Hafiz and Al-Goria established a novel family of optimal eighth order iterative algorithms and then studied their dynamics. In [23], the authors introduced seventh and ninth orders novel iterative algorithms with the help of the predictor-corrector technique and Simpson quadrature formulae. By employing the Newton interpolation technique along with weight functions, Salimi et al. [24] introduced a new family of eighth order optimal root-finding algorithms. In [25], the authors constructed some novel optimal iterative algorithms with higher convergence and demonstrated the applicability of the suggested methods by solving some engineering problems. Recently, Chu et al. [26] proposed a novel family iteration scheme and discussed the dynamics of the presented methods with the help of computer tools.In the present research article, we introduce a new fourth-order and derivative free algorithm for solving engineering problems in the form of scalar nonlinear functions. The construction of this algorithm is based upon the finite difference scheme on Ostrowski’s method. We also certify that the designed algorithm has fourth-order convergence. The designed algorithm is then applied to some real-world engineering problems for certifying its better performance and applicability among the other fourth-order algorithms in the literature. The dynamical comparison of the designed algorithm with the other comparable ones has been also presented via the computer program Mathematica 12.0.


## 2. Main Results

Consider the nonlinear problem of the following form:(1)ψu=0,where ψ is a real-valued function with an open interval domain.Suppose thatα is a root of (1) with u0 as an initial guess near to the exact root α, then the implication of Taylor’s series around u0 for (1) gives us(2)ψu0−u−u0ψ′u0+u−u02ψ′′u02!+⋯=0.Ifψ′u0 is nonzero, then the above expression implies(3)ui+1=ui−ψuiψ′ui,which is Newton’s root-finding algorithm [1, 2] for scalar nonlinear functions.By taking it as a predictor, Ostrowski designed the following two-step iterative algorithm:(4)vi=ui−ψuiψ′ui,ui+1=vi−ψviψuiψ′uiψui−2ψvi,which is well-known Ostrowski’s root-finding algorithm [11] for scalar nonlinear functions.By including Newton’s algorithm, the above two-step method may be converted to three-step in the following form:(5)vi=ui−ψuiψ′ui,wi=vi−ψviψ′vi,ui+1=wi−ψwiψviψ′viψvi−2ψwi,which is a three-step iteration scheme for calculating zeros of nonlinear scalar equations. The main drawback of the above algorithm is its high computational cost per iteration as it requires six evaluations for its execution. To lower its computational cost make it more effective, we approximate its first derivatives and make it derivative free, so that it can be easily applied on those nonlinear scalar functions whose first derivative becomes infinite or does not exist. To approximate ψ′u in the predictor step, we employ the forward difference approximation as(6)ψ′ui=ψui+ψuiψui=gui.To approximateψ′v, we utilize the finite difference scheme as(7)ψ′vi=ψvi−ψuivi−ui=hui,vi.Using (6) and (7) in (5), we can write Algorithm 1.Algorithm 1.
For a givenu0, compute the approximate solution ui+1 by the following iterative schemesvi=ui−ψui/gui,i=0,1,2,…,wi=vi−ψvi/hui,vi,ui+1=wi−ψwiψvi/hui,viψvi−2ψwiAlgorithm 1 is a new iteration scheme for calculating the approximated roots of scalar nonlinear equations and needs only four evaluations per iteration. The main characteristic of the suggested algorithm is that it is derivative free and easily applicable to all those scalar functions whose derivatives become undefined within the domain. In this sense, the proposed algorithm’s computing cost is minimal which results in a higher efficiency index.


## 3. Convergence Analysis

In the present section, we shall discuss the convergence criterion of the newly designed algorithm, i.e., Algorithm 1.Table 1
Comparison among different fourth-order algorithms.
MethodsITui+1ψui+1σ=ui+1−uiηOM040.34264820581144999.186801e−172.728697e−054TM050.34264820581144994.236557e−231.848727e−144ZM050.34264820581144997.675869e−251.516654e−074Algorithm 1040.34264820581144995.362719e−181.131404e−064Theorem 1.
Suppose thatα is the root of the equation ψu=0. If ψu is sufficiently smooth in the neighborhood of α, then the order of convergence of Algorithm 1 is at least four.Proof.
To analyze the convergence criterion of the iteration scheme (1), we assume that α is a root of equation ψu=0 and ei be the error at ith iteration; then, ei=ui−α, and by using Taylor’s series expansion, we have(8)ψui=ψ′αei+12!ψ″αei2+13!ψ″′αei3+14!ψivαei4+Oei5,ψui=ψ′αei+d2ei2+d3ei3+d4ei4+Oei5,where(9)di=1i!ψiαψ′α,gui=ψ′α1+3d2ei+7d3+d22ei2+6d2d3+15d24ei3+18d2d4+31d5+d3d22+5d32ei4+Oei5.
With the help of equations (8) and (9), we get(10)vi=α+2d2ei2+6d3−5d22ei3+14d4−26d3d2+13d23ei4+Oei5,(11)ψvi=ψ′α2d2ei2+6d3−5d22ei3+14d4−26d3d2+13d23ei4+Oei5,(12)hui,vi=ψ′α1+d2ei+d3+d22ei2+8d2d3−5d23+d4ei3+13d24−27d3d22+16d4d2++d5+6d32ei4+Oei5,(13)wi=α+2d2ei3+8d2d3−7d23ei4+Oei5,(14)ψwi=ψ′α2d2ei3+8d2d3−7d23ei4+Oei5.
Using equations (8)–(14) in Algorithm 1 gives us the following equality:(15)ui+1=α−2d23ei4+Oe5,which implies that(16)ei+1=−2d23ei4+Oe5.
The above equation shows that the designed algorithm is of fourth-order convergence.


## 4. Real-Life Applications

In this section, we take five real-world problems in the form of scalar nonlinear functions to exhibit the applicability, validity, and efficiency of the newly designed fourth-order algorithm. We compare it with other well-known fourth-order algorithms, namely, Ostrowski’s method (OM) [11], Traub’s method (TM) [12], and Zhanlav method (ZM) [27].Example 1.
Fluid permeability problem:
The hydraulic permeability is actually the measurement of the flow resistance. It relates the pressure gradient to fluid velocity and may be expressed as(17)κ=reu3201−u2,reu3−20k1−u2=0,where κ denotes the specific hydraulic permeability, re stands for the radius, and 0≤u≤1 is the porosity. For further details see [28] and the reference cited therein. By taking the values of re=100 and κ=0.4655 in (14), we obtain the above problem in the following nonlinear function:(18)ψ1u=100u3−9.311−u2.
To solveψ1, the initial guess has been chosen as u0=2.0 for starting the iteration process, and the results are given in Table 1.Table 2
Comparison among different fourth-order algorithms.
MethodsITui+1ψui+1σ=ui+1−uiηOM030.08643355805229182.960661e−184.952936e−054TM030.08643355805229163.697508e−161.642942e−044ZM120.08643355805229181.957052e−255.571776e−074Algorithm 1030.08643355805229171.940272e−173.210544e−064Example 2.
Blood rheology model.
Blood rheology is a branch of science that works to study the physical and flow properties of blood [29]. Blood is actually a non-Newtonian fluid and treated as Caisson fluid. The 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(19)H=1−167u+43u−121u4,where flow rate reduction is computed by H. Using H=0.40 in (19), we have(20)ψ2u=1441u8−863u5−0.05714285714u4+169u2−3.624489796u+0.3.
To solveψ2, the initial guess has been chosen as u0=0.9 for starting the iteration process, and the results are given in Table 2.Table 3
Comparison among different fourth-order algorithms.
MethodsITui+1ψui+1σ=ui+1−uiηOM091.92984624284786221.626245e−221.858593e−064TM101.92984624284786224.578052e−271.235520e−074ZM441.92984624284786221.365062e−196.456152e−064Algorithm 1061.92984624284786221.335843e−293.841021e−104Example 3.
Van Der Wall’s equation.
The well-known equation for examining the behaviour of real and ideal gas was introduced by Van Der Wall’s [30], with the following expression:(21)P+C1n2V2V−nC2=iRT.
Equation (21) may be easily transformed into the following nonlinear function by taking the particular values of the parameters:(22)ψ3u=0.986u3−5.181u2+9.067u−5.289,where u is the gas volume that may be easily determined by solving ψ2. Because the polynomial’s degree is three, it must have three roots. There is only one positive real root 1.9298462428 among these which is physically possible since the gas volume can never be negative. To solve ψ3, the initial guess has been chosen as u0=1.0 for starting the iteration process, and the results are given in Table 3.Table 4
Comparison among different fourth-order algorithms.
MethodsITui+1ψui+1σ=ui+1−uiηOM04−0.00000000000000001.030994e−513.261218e−134TM04−0.00000000000000001.706136e−361.719178e−094ZM04−0.00000000000000001.383001e−201.153483e−054Algorithm 104−0.00000000000000002.759067e−456.858899e−144Example 4.
Plank’s radiation law.
The energy density within the black isothermal body is calculated using Planck’s radiation law [31] given as follows:(23)φγ=8πPcγ5ePc/γTk−1.
Suppose we want to calculate wavelengthγ1 for the peak value of the energy density φγ1. To transform (23) in nonlinear form, we assume u=Pc/γTk and obtain the following nonlinear expression:(24)ψ4u=−1+u5+e−u.
One of the estimated roots ofψ4 is −0.0000000000000000 which represents the maximum amount of the wavelength of the radiation. To solve ψ4, the initial point has been chosen as u0=−2.0 for starting the iteration process, and the results are given in Table 4.Table 5
Comparison of different fourth-order algorithms.
MethodsITui+1ψui+1σ=ui+1−uiηOM032.00211877895382731.247083e−312.494230e−084TM032.00211877895382731.271632e−297.400791e−084ZM032.00211877895382731.334997e−215.297174e−064Algorithm 1032.00211877895382731.760330e−281.073092e−094Example 5.
The problem of beam designing.
In Physics and Engineering sciences, the beam designing problem [32] regarding the embedment u of a sheet pile wall in the form of scalar nonlinear function is expressed as(25)ψ5u=u3+2.87u2−4.62u−10.284.62.
To solveψ5, the initial guess has been chosen as u0=3.0 for starting the iteration process, and the results are given in Table 5.
Here, we choose the accuracyε=10−15 in the following stopping criterion of the computer program:(26)ui+1−ui<ε.
We used the computer application Maple 13 to solve all numerical problems.
Tables1–5 exhibit the numerical comparison of the designed fourth-order algorithm with Ostrowski’s method (OM), Traub’s method (TM), and Zhanlav’s method (ZM). In the columns of the above tables, ITstands for the number of iterations, ψu indicates the positive value of the function ψu, ui+1indicates the estimated root, σ indicates the absolute difference of the consecutive estimations ui+1, and ui and η represent the approximated computational order of convergence given as(27)η≈Inui+1−α/ui−αlnui−α/ui−1−α,which was introduced by Weerakoon and Fernando [33].


## 5. Dynamical Analysis via Computer Technology

In this section, we give a detailed graphical comparison of the newly designed fourth-order algorithm with the other fourth-order algorithms via computer technology by considering some complex polynomials in the form of polynomiographs. A polynomiograph is a graphical object generated in a process known as polynomiography, introduced by Dr. Bahman Kalantri in 2005 [34]. It is defined as “the algorithmic visualization of polynomial equations by employing different iterative techniques” [35].To draw dynamics by employing computer technology using various iterative algorithms, an initial rectangleℛ which includes the root of the investigated complex polynomial has been chosen. Then, for every point w0 in ℛ, we perform the process of iteration. The image’s quality is usually correlated with the discretization ofℛ, i.e., if the rectangle ℛ has been discretized into a 2000×2000 grid, then the quality of the produced image will be better.Typically, the colors of produced polynomiographs are fully associated with the number of iterations required to find the approximated roots with a given precision and a selected iterative algorithm. The main algorithm for the production of a polynomiograph is given in Algorithm1.Algorithm 1: Polynomiograph’s generation.
Input:q∈ℂ—polynomial, A⊂ℂ—area, K—maximum No. of iterations, I—iterative algorithm, ε—accuracy, colormap 0…C−1—colormap with C colorsOutput: polynomiograph for the complex polynomial q in the area Aforw0∈A, doi=0whilei≤K dowi+1=Iwiifwi+1−wi<∈, thenbreaki=i+1colorw0 via colormapA stopping criterion is always required for an iterative algorithm that includes the repetition of steps, since it informs us about the convergence or divergence of the investigated iterative algorithm. Such a criterion is commonly referred to as a convergence test with the following mathematical expression:(28)wi+1−wi<ε,where wi+1 and wi denote the successive iterations, and ε >0 stands for the accuracy in the stopping criterion. The convergence test wi+1,wi,ε is considered TRUE if the iterative algorithm under consideration is converged and FALSE if it is diverged. The abovementioned stopping criterion (28) is also used in this study. The variety of polynomiographs’ colors is correlated with the performed iterations to find out the root with given precision ε. Using various iterative algorithms, a variety of aesthetically pleasant polynomiographs can be produced by altering the parameter K, where K specifies the upper limit of the number of iterations. For further information regarding polynomiography along with its applications in different fields, one can see [36–44] and the references cited therein.For drawing polynomiographs through different iterative algorithms, we consider the following four complex polynomials:(29)q1w=w3−1,q2w=w3−12,q3w=w4−1,q4w=w4−12.The colormap used for the coloring of iterations in the generation of polynomiographs is shown in Figure1:Figure 1
The colormap used for generating polynomiographs.Example 6.
Polynomiographs for the polynomialq1 through different fourth-order algorithms.
In the first example, we consider a cubic-degree polynomialq1w=w3−1, having three distinct roots 1,−1/2+3/2i, and −1/2−3/2i.We used a computer program to run all the methods to get the simple roots of the under consideration polynomial q1, and the results are shown in Figure 2.Figure 2
Polynomiographs related to the complex polynomialq1. (a) Ostrowski’s method. (b) Traub’s method. (c) Zhanlav’s method. (d) Algorithm 1.
(a)(b)(c)(d)Example 7.
Polynomiographs for the polynomialq2 through different fourth-order algorithms.
In the second example, we take a sextic-degree polynomialq2w=w3−12, which has three unique roots 1,−1/2+3/2i, and −1/2−3/2i with multiplicity two. We perform the process of iteration for all iterative algorithms for drawing polynomiographs, and the results are shown in Figure 3.Figure 3
Polynomiographs related to the complex polynomialq2. (a) Ostrowski’s method. (b) Traub’s method. (c) Zhanlav’s method. (d) Algorithm 1.
(a)(b)(c)(d)Example 8.
Polynomiographs for the polynomialq3 through different fourth-order algorithms.
In the third example, we consider a quartic-degree polynomialq3w=w4−1, which has four unique roots 1,−1,i,and−i. We created the graphical objects by executing all iterative algorithms, and the results are shown in Figure 4.Figure 4
Polynomiographs related to the complex polynomialq3. (a) Ostrowski’s method. (b) Traub’s method. (c) Zhanlav’s method. (d) Algorithm 1.
(a)(b)(c)(d)Example 9.
Polynomiographs for the polynomialq4 through different fourth-order algorithms.
In the fourth example we take an eighth-degree complex polynomialq4w=w4−12 with four unique roots 1,−1,i,and−i of multiplicity two. We used a computer program to run all methods for drawing polynomiographs, and the results in the form of visually attractive pictures are shown in Figure 5.
In above examples, we compared the developed algorithm to various fourth-order iterative algorithms using a computer program by taking into account different degrees complex polynomials. Two key features may be identified from the produced graphics. The first is the iteration scheme’s speed of convergence, and the second feature is the iteration scheme’s dynamics. Low dynamics are seen in places with little color variation, and high dynamics are found in areas with a lot of color variety. The black coloring in the graphics denotes areas where the solution cannot be reached in the specified number of iterations. The darker zone in the above-presented pictures indicates that the iterative algorithm under consideration requires fewer iterations for finding the solution of the given problem. The same-colored regions in the graphical objects represent the same number of iterations necessary to find the required solution with the given accuracy. Note that the polynomiographs created using our proposed iterative algorithm have considerably brighter and darker regions and no black areas as compared to other similar order algorithms in the literature. Furthermore, the polynomiographs of the proposed iterative algorithm show larger convergence areas than the other comparable techniques which demonstrate the better efficiency of the suggested algorithm.
We drew all graphical objects with the computer program Mathematica12.0 by using the values of parameters as ε=0.001 and K=20, where ε and K indicate the accuracy and the upper bound of the number of iterations, respectively.Figure 5
Polynomiographs related to the complex polynomialq4. (a) Ostrowski’s method. (b) Traub’s method. (c) Zhanlav’s method. (d) Algorithm 1.
(a)(b)(c)(d)


## 6. Conclusion

By employing the finite difference scheme on Ostrowski’s method, we designed a new derivative free algorithm for calculating the approximate zeros of nonlinear scalar equations that possesses the fourth-order convergence. To analyze the applicability of the designed algorithm, we took some real-life engineering problems and solved them via computer tools. The numerical results given in Tables1–5 proved the better performance and applicability of the designed algorithm against the other fourth-order algorithms. We have also presented the dynamics of the designed algorithm and gave a detailed comparison with the other comparable fourth-order algorithms in the literature via computer tools that revealed the convergence and other graphical characteristics of the designed algorithm. A new family of derivative free root-finding algorithms can be constructed by applying the finite difference scheme to the existing methods in the literature.



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