Modification of Behl ' s Method Free from Second Derivative with an Optimal Order of Convergence

Behl’s method is one of the iterative methods to solve a nonlinear equation that converges cubically. In this paper, we modified the iterative method with real parameter β using second Taylor’s series expansion and reduce the second derivative of the proposed method using the equality of Chun-Kim and Newton Steffensen. The result showed that the proposed method has a fourth-order convergence for  = 0 and involves three evaluation functions per iteration with the efficiency index equal to 41/3  1.5874. Numerical simulation is presented for several functions to demonstrate the performance of the new method. The final results show that the proposed method has better performance as compared to some other iterative methods.


INTRODUCTION
Mathematical modeling is a mathematical representation of sciences and engineering problems. One of the terms of mathematical representation is a nonlinear equation. While in fact, most of the nonlinear equation consists of complexity form of function [1]. The problem arises when we solve this equation: (1) Most of the complicated nonlinear equations can't be solved using the analytical technique. So, numerical solving is an alternative solution using requiring calculating. This technique is most well known as an iterative method. The iterative method that are popular to find the root of (1) is the Newton Method i.e.
Equation (2) is a one-point iterative method that is constructed by Taylor expansion in first order with quadratic order of convergence. This method involved two evaluation functions with efficiency index is 2 1/2  1.4142. Furthermore, to fix up the order of convergence, second-order Taylor expansion is used to construct an iterative method. It processes appear some one-point iterative method with third-order of convergence, such as Halley's, Halley's Irrational, and Chebyshev's methods [2]. Besides that, some authors develop one-point iterative method with third-order of convergence using several technique approximation, such as quadratic function [3] [4], hyperbolic function [5], parabolic function [6], Adomian decomposition method [7] [8], homotopy perturbation method [7], curvature [9], variation iteration method [10], and substituted Taylor series [11] [12]. All of the third-order iterative methods involve three functional evaluations with index efficiency equal to 3 1/3  1.4422.
The one-point iterative methods above have third-order of convergence. So, based on Kung-Traub conjecture [13], the iterative methods aren't an optimal order of convergence. To improve the convergence order of an iterative method, some author uses several approximations. One of the approaches used is the substitution of the Taylor second-order series as has been done by Chun and Kim [9], Wartono and Nanda [12], and Baghat [11].
In this paper, we construct a new iterative method using a combination of a one-point iterative method developed by Behl et al. [14] and second-order Taylor series. Furthermore, at the end of this section, we give numerical simulation to examine the performance of our proposed method and some iterative methods. The performance will be compared based on the number of iterations, computational of the order of convergence, the absolute value of a function, absolute error and relative error.

METHOD
In this section, we use some definitions and theorems that are used to construct an iterative method, finding for the order of convergence both of using Taylor series or computational order of convergence, and numerical simulation. Theorem 1. [15] Let f has (n +1) continuous derivative on [a, b] for some n  0, and let x, x0  [a, b]. Then there exists a point  between x0 and x such that be a real function with simple root  and let    ( The relation in Equation (3) called as an error equation in the (n + 1) th .

Definition 3. [1]
Let d be the number of any evaluation of a function or one of its derivatives. The efficiency of the method is measured by the concept of efficiency index and is defined by where p is the convergence order of the method. Definition 4. [1] Suppose that xn  1, xn and xn + 1 are three successive closer to the root . Then the computational order of convergence  is approximated by

The Developed Iterative Method
In this sub-section, the author constructs an iterative method by considering a one-point iterative method developed by Behl et al. [14] and add one real parameter  as follows: Equation (4) is a third-order iterative method with three functional evaluations and an efficiency index as 1.4422. Furthermore, to modify (4), we consider a second-order Taylor series expansion about n x written as, , we can write (5) as: The term n n x x  1 at the right of (6) will be substituted by (4). Now, we can rewrite (6) as follows To fix up the order of convergence and avoid using the second derivative of f(x), we approach ) ( " n x f using the equality of two third-order iterative methods. We consider Chun-Kim's [9] and Newton-Steffensen's methods [4], respectively, written as follows: and where Both of iterative method of (8) or (9) have third order of convergence. Based on these equations, will be approximated using equality of (8) and (9) as Wartono et al. [16] which is written as Using simplification, we obtain a form of f ''(xn) as follows Furthermore, by substituting (10) into (7) we obtain an equation to iterative method Equation (11) is a two-point iterative method involving two functions and one of its first derivative.

Convergence Analysis
The theorem 2 describes the order of convergence in (11) So, using n n x e    , Equation (12) can be written as where . Based on (13) and (14) From (13) and (20), we obtain the multiple of f(xn) and f(yn) as follows Applying (15), (29), and (30) in (11), and then simplify it, we obtain Substitute (31) into (11), and using , we get a convergence order of the proposed method in (11) that is given by Equation (32) gives us information that the order of convergence of the proposed method in (11)

Numerical Simulation
In this section, we present a numerical simulation to show the performance of the proposed method for 0   (MMIOT), and then we compare the proposed method between some kind of iterative methods, namely: Newton's method (MN), third-order iterative method (MIOT), Newton-Steffensen's (MNS) [4] and Chun-Kim's method (MCK) [8]. Some measures of performance of the compared methods are the number of iterations (IT), computational order of convergence (COC), and the absolute value of a function at n th iteration ( |f(xn)| ).
We used several following real test functions and all computations have been performed using MAPLE 13 with 800 digits floating-point arithmetic. The root of real test functions is displayed by the computed approximate zeros α round up to 20th decimal places.
The number of iteration (IT) for the compared methods are determined using the following stopping criteria , where ε is the precision of the compared methods, while the COC is approximated by the formula Besides using the Taylor expansion approach, we also use computational approximation using formulation in (33) to get the convergence order of the compared iterative method. The results as seen in Table 2. Table 2 describes that the computational order of convergence of the proposed method (MMIOT) is four. In addition to using the number of iterations and COCs, the performance of an iterative method can also be measured using accuracy, namely calculating the absolute of a function as given in Table 3. Table 3 shows the absolute value of the function using a total number of functional evaluations (TNFE), which are as many as twelve functional evaluations. Based on Table  3, it can be seen that the proposed method has better accuracy than other iterative methods.