Three-point iterative algorithm in the absence of the derivative for solving nonlinear equations and their basins of attraction

In this paper, we suggested and analyzed a new higher-order iterative algorithm for solving nonlinear equation g(x)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(x)=0$$\end{document}, g:R⟶R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g:{\mathbb {R}}\longrightarrow {\mathbb {R}}$$\end{document}, which is free from derivative by using the approximate version of the first derivative, and we studied the basins of attraction for the proposed iterative algorithm to find complex roots of complex functions g:C⟶C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g:{\mathbb {C}}\longrightarrow {\mathbb {C}}$$\end{document}. To show the effectiveness of the proposed algorithm for the real and the complex domains, the numerical results for the considered examples are given and graphically clarified. The basins of attraction of the existing methods and our algorithm are offered and compared to clarify their performance. The proposed algorithm satisfied the condition such that |xm-α|<1.0×10-15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|x_{m}-\alpha |<1.0 \times 10^{-15}$$\end{document}, as well as the maximum number of iterations is less than or equal to 3, so the proposed algorithm can be applied to efficiently solve numerous type non-linear equations.

most powerful tool to find the roots of nonlinear equation, boundary value problem and system of nonlinear equations etc. The maximum attainable computational efficiency of multi-point without method is closely related to the hypothesis given by Kung and Traub [1] and had conjecture that the convergence order of any multipoint method without memory with n-evaluation is not larger than 2 (n−1) . A number of modification of Newton's method with improved rate of convergence are reported by previous researcher and there in. Some scheme developed from Newton method by some authors are given. Many papers are written about iterative methods for solving the non-linear equations for details, see [2][3][4][5]. Proposed and analyzed three new root-finding algorithms for solving nonlinear equations in one variable and derived these algorithms with the help of variational iteration technique, see for instance [6]. The variant of Frontini-Sormani method, some higher order methods for finding the roots (simple and multiple) of nonlinear equations are proposed. In particular, and have constructed an optimal fourth order method and a family of sixth order method for finding a simple root (see for instance, [7,8]). The basin of attraction is a method to visually sense how an approach makes as a function of the different starting points. In this work, we discuss the possibility of approximating the derivative by suitable difference approximations. It is shown that the presented algorithm convergence eighth order and this theory is supported by computational results. It is observed that for several functions, suggested algorithm can produce even better accuracy than that of other methods, we think about an iterative method for solving non-linear equations in real and complex domains, which are a significant zone of research in the numerical analysis as it has intriguing applications in several branches of pure and applied science can be concentrated in the overall of the non-linear equations, for getting a simple root α of the function g : R −→ R i.e. g(α) = 0 , and g ′ (α) � = 0 , we know the method of Newton for finding α utilized the iterative method The method of Newton is the most popular and simple algorithm, which incorporates the derivative of the function. However, Steffensen's method [9,10] a variety of Newton's technique which doesn't utilize the derivative of the function. In this technique, the derivative is approximated numerically by the central difference scheme. Steffensen's technique has the same order of convergence as Newton's method, based on the approximation of the first derivative. The motivation behind this work is to improve a new eight-order derivative-free algorithm. This work is ordered as follows. In the "Preliminaries" section, basic concepts used in the work are presented. The purpose of the "Construction of presented iterative method and analysis of convergence" section studies the construction of the proposed method and analyses the convergence order of the proposed method. While the "Results and discussion" section presents results and discussion in real and complex domains, in the "Numerical problems in real domain" section, we consider five numerical examples to demonstrate the performance of the proposed algorithm and in the "Graphical comparison for the basins of attraction" section Graphical comparison by Means of the basins of attractions. "Some real-life applications" section, four application problems are solved. Finally the "Conclusion and future work" section concludes the paper.

Preliminaries
Suppose that g(x)=0, x ∈ R, is called nonlinear equation if the function g(x) is explicitly algebraic function as polynomial of degree other than one or a transcendental function of x and they do not involve derivative or integral. A value for parameter x that satisfies the equation g(x)=0 is called a root or a zero of g(x). The accompanying significant definitions given underneath are needed for the ensuing convergence analysis.
Definition 1 [11] Suppose that g : [a, b] → R . Allow the next conditions to hold then the succession {x m } defined by Newton's method beginning with an initial estimate Additionally, we have the next assessment holds, where C 1 = max x∈[a,b] |g ′′ (x)|. and C 2 = min x∈[a,b] |g ′ (x)|.
Definition 2 [12] Let a real function g(x) with a root α and suppose that {x m } be a sequence of real numbers which converge towards α . The order of convergence p is given by where ξ is constant called the asymptotic error and p ∈ R + . Definition 3 [13] Suppose that e m = x m − α is m th iteration error, then the equation error is If the error of equation exists, then p is convergence order of an iterative method.
Definition 4 [13] Let x m−1 , x m and x m+1 are three iterations closer to α . The computational order of convergence might be approximated by . Definition 5 [4] Let a number α and its an approximation α c . We will think about two different ways to calculate the error in such an approximation Absolute Error = |α c − α| and Relative Error = |α c − α||α| −1 . Also in this work we will discuss some specific problems using the basin of attraction as a standard for comparison. Now, we shall requisition some definitions, see in [14]. Let R : C −→ C , is a rational map on Riemann sphere.

Definition 7
Let z 0 is a starting point of rational map if R(z 0 ) = z 0 .

Definition 8
Let z 0 is a periodic point with period m which is such that R m (z 0 ) = z 0 where m is the smallest such integer.
If the derivative is also zero then the point is called super-attracting.

Construction of presented iterative method and analysis of convergence
For solving nonlinear equations, we drive the derivative-free iterative technique by using the approximate version of the first derivative of g ′ (x m ) by where θ ∈ R and not equal zero. Let us consider the method in [15]: By using Eq. (1), we obtain the following new eighth order algorithm in the absence of the derivative which using for solving a nonlinear equation as follows.
Eighth order derivative free iteration algorithm (8th BM): Further, we substitute the approximation of the derivative g ′ (x) in Eq. (2) by Eq. (1), we get the proposed algorithm free from derivatives, as follows: 8th BM: Given an initial approximation x 0 (close to α ) the root of g(x) = 0 . We find the approximate solution . Steps for calculating root using 8th BM Step 1: Define the function g(x).
Step 4: We use a specific accuracy ǫ as |x i − α| < ǫ , and repeat Step 3 until we get desired approximate root which satisfy the condition. In order to prove the convergence of 8th BM, we establish the following theorem with the help of Maple software.
Theorem Suppose that g(x) : R −→ R for the interval (a, b). Assume that g(x) has sufficiently continuous derivatives in (a, b). If α has a simple root of g(x) and if x 0 is closed to α then 8th BM satisfies the following error equation:

Proof
Let the error at step m be denoted by e m = x m − α and F = g ′ (α) and Computing g 2 (x m ) using Eq. (5), then multiply by 2θ we get Expand g(x m + θg(x m )) and g(x m − θ g(x m )) around the root α and express it in terms of powers of error e m we get , 3 2 )e 4 m + ... By using Eqs. (13) and (15)  The last equation shows that 8th BM is eight order of convergence. This completes the proof.

Numerical problems in real domain
In this section, we give the results of some numerical examples to compare our proposed algorithm with the methods in [16] which are called Dehghan Method 2 (DM2), King Method (KM) and Proposed Free Derivative Method (PFDM). We are using five examples to display the effectiveness of the presented algorithm. All the computations were done by using Maple 18 and were satisfied the condition such that |x m − α| < 1.0 × 10 −15 , as well as the maximum number of tainiterations is less than or equal to three. The computational results in Table 1 lists the absolute value of the given nonlinear function g i (x m ), i = 1, 2, 3, 4, 5, m = 3, for our proposed algorithm at θ = 1, −1, 0.5, −0.5 . In addition, it can be seen that in Table 1 the computational order of convergence (COC) perfectly coincides with the theoretical results. The results are given in Table 1 in terms of the number significant digits for each test function at 3rd iteration, that is, e.g. 1.0 × 10 −41 shows that the absolute value of the given nonlinear function g 1 (x 3 ) at 3rd iteration is zero up to 41 decimal places. In Table 2, "Div" indicates that the algorithm does not converge after the maximum allowed iteration is reached. From Table 2       DM2 requires 4 iterations. So the quickest algorithm to hit the root is ours. In g 2 (x) the method having the least iteration is 8th BM. As far as the numerical results are concerned, for most of the functions we tested, the proposed algorithm can be competitive with the methods we are comparing. The computational results presented in Table 1 and Table 2 show that our algorithm is more efficient compared with the proposed methods in [16].

Graphical comparison for the basins of attraction
Here we examine the comparison of some high order simple root finder in the complex plane using a basin of attraction. We consider the polynomial g(z) = z r − 1; z ∈ C for achieving the unity roots in the form The basin of attraction compared to the roots of the function g(z) comprises of all beginning points z 0 which are pulled to ω k . We use a comparison between iterative methods by using these basins. In the computational examples, let D = [−2, 2] × [−2, 2] ∈ C of 250 × 250 points, furthermore, we apply our algorithm beginning in each a z 0 in D. The basin of attraction for complex Newton's method was first started by [13]. The basin of attraction is an approach to see how a calculation acts as a function for the different beginning points. It is another approach to look at the iterative methods. We give a color for each point z 0 ∈ C according to the root at which the corresponding iterative algorithm starting from z 0 converges, for details, one may see [17,18]. the accompanying test functions had been considered of comparison: g(z) = z r − 1, r = 2, 3, 4, 5 respectively. We compare the newly proposed algorithm, namely (8th BM) and four different methods as Bhavna Panday and Jai Prakash Jaiswa [13], Changbum method (CMB) [19], Sharma Methods (SB) [20] and Behzad method (BG) [21]. We choose nonlinear functions to provide the accuracy of the newly proposed algorithm for different θ to find complex roots for complex functions. The roots of used functions are listed and the computations reported using Maple 18 had been done. More scientific calculations in numerous territories of science request a high exactness level of numerical accuracy. We use the next applications for the comparison of the other methods as follows ω k = cos 2π(k − 1) r + i sin 2π(k − 1) r ; k = 1, 2, ..., r. Fig. 6 Plots of 8th BM for θ = 0.5 , θ = 1 and the method in [13], respectively for g 1 (z).

Fig. 7
Plots of 8th BM for θ = 0.5 , θ = 1 and the method in SB [20] respectively for g 2 (z). The sequence {z k } ∞ k=0 is of the point orbit, if this converges to the root then we say that is attracted to. the initial points for the sequence of converges to z * is the basin of Fig. 8 Plots of 8th BM for θ = 0.5 , θ = 1 and the method in BG [21] respectively for g 3 (z).

Fig. 10
Plots of 8th BM for θ = 0.5 and θ = 1 respectively for g 5 (z). attraction of. Boundaries between basins generally are fractals in nature. 8th BM which is given for real domain is also used to achieve the graphs of complex polynomials that envision the roots getting process. Figures 6 and 7 show the basins of attraction of 8th BM at θ = 0.5 , θ = 1 and the methods in [13,21], from left to right respectively for the quadratic and cubic polynomials. the red of color shows the roots z * . This shows that the convergence of 8th BM when the initial points are chosen near the root is rapid convergence, as the red intensity of the colors emphasizes that the proposed algorithm converges in less than 5 iterations. The 8th BM is more accurate with few iterations number and most basins of attraction at θ = 0.5 , θ = 1 for g 1 (z) . Figures 8, 9, 10, 11 and 12 show the basins of attractions of 8th BM and the other methods in [13,15,22], where the presented algorithm is globally convergent with the lowest iterations number. when the polynomial degree increments from 3 to 7, the 8th BM has difficulties, and their iterations number increments. 8th BM has small spread points compared with the others.

Some real-life applications
In this section we present some applications and compare our results to well-known methods: Fig. 11 Plots of 8th BM for θ = 0.5 and θ = 1 and the method in [13] respectively for g 6 (z).

Fig. 12
Plots of 8th BM for θ = 0.5 and θ = 1 respectively for g 7 (z) and the method in [13] Page 14 of 17 Bahgat J Egypt Math Soc (2021) 29:23 Application 1 The deepness of embedment x if a sheet-pile wall is governed by the following equation [23]: It can be rewritten as An engineer has rated the deepness to be x = 2.5 . Here we get the root of the equation g(x) = 0 with initial point 2.5 and compare some fully famous methods to our proposed algorithm.

Application 2
The vertical stress η z created at point in an elastic continuum under the brink of a strip base supporting a regular pressure p is given via Boussinesq's formula [23] to be: A scientist is interested in estimating the value of x at which the vertical stress η z ought to be 25 percent of the footing stress p. Initially it is rated that x = 0.4 . The above can be rewritten for η z being equal to 25 percent of the footing stress p : Now we find the root of the equation g(x) = 0 with initial point 0.4 and compare some well famous methods to our proposed algorithm.

Application 3
In general, many applications in science and engineering which include definition of unknown in turn lead to root-finding problem. The Planck's radiance law problem appearing in [24,25] is one among them and it is given by which calculates the density of energy during an isothermal blackbody. Here, µ is the wavelength of the radiation; T is the absolute temperature of the blackbody; k is Boltzmann's constant; h is the Planck's constant; and c is the speed of light. assume that we would like to define wavelength µ , which corresponds to maximum the density of energy φ(µ  Let us define The aim is to find a root of the equation g(x) = 0 . Obviously, one of the root x = 0 is not taken for discussion. As argued in [24], g(x) = 0 for x = 5 and e −5 ≈ 6.74 × 10 −3 . Hence, it is expected that another root of the equation g(x) = 0 might occur near x = 5 . The approximate root of g(x) is given by 4.96511423174427630369. Consequently, the wavelength of radiation µ corresponding to which the energy density is maximum is approximated as µ ≈ hc KT 4.96511423174427630369.
Application 4 Study of the multipactor effect [26]. The trajectory of an electron in the air gap between two parallel plates is given by where E 0 sin(ωt + �) is the RF electric field between plates at time t 0 , x 0 and ν 0 are the position and velocity of the electron e and m are the charge and mass of the electron at rest respectively. For the particular parameters, one can deal with a simpler expression as follows: The required zero of the above function is −0.3094661392082146514.... Table 3 show the numerical calculations with respect to iterations number (m). The numerical applications of the above real life experiments demonstrate the validity and applicability of the proposed algorithm. This shows that the proposed algorithm is very much appropriate for all the application experiments . In most of the cases, the proposed algorithm show better performance in comparison to the existent methods.

Conclusion and future work
In this study, we suggested a derivative-free iterative algorithm with different values of the parameter θ to solve nonlinear equations in real and complex domains. Considering that the proposed algorithm is derivative-free this allows us to apply it also on nonsmooth equations with positive and promising results. Moreover, this algorithm is particularly appropriate, to those applications in which the required derivatives are lengthy . Tables 1 and 2 display the best performance of the suggested algorithm in terms of accuracy, speed, number of iterations, and computational order of convergence as compared to other known algorithms. Figures 1-5 show that 8th BM converges quickly and more accurately at least the number of iterations. Figures 6-12 show that the basins of attraction of the new algorithm known can compete with other optimal eighth order algorithms in the literature. Theoretical and COC are verified in the considered problems. Five examples in the real domain and seven in the complex domain are solved where 8th BM produces better results than compared methods. The maximum number of iterations is less than or equal to three, to reach an absolute error less than 10 −15 . Four real life applications are solved where the new algorithm produce better results than other compared methods.
In the upcoming future we plan to progress as follows. We will research solution of systems with large number of equations. Also, We will improve the codes so it handle a system of algebraic equations. Abbreviations 8th BM:: Eighth order derivative free iteration algorithm..