EFFICIENT SIXTH ORDER ITERATIVE METHOD FREE FROM HIGHER DERIVATIVES FOR NONLINEAR EQUATIONS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we proposed new iterative sixth order convergence method for solving nonlinear equations. The combination of the Taylor series and composition approach is used to derive the new method. Numerous methods have been developed by many researchers whenever the function’s second and higher order derivatives exist in the neighbourhood of the root. Computing the second and higher derivative of a function is a very cumbersome and time consuming task. In terms of low computation cost, the newly proposed method finds the best approximation to the root of non-linear equations by evaluating the function and its first derivative. The proposed method has been theoretically demonstrated to have sixth-order convergence. The proposed method has an efficiency index of 1.56. Several comparisons of the proposed method with the various existing iterative method of the same order have been performed on the number of problems. Finally, the computational results suggest that the newly proposed method is efficient compared to the well-known existing methods.


INTRODUCTION
The solution of nonlinear equations governing the natural phenomena of real word problems is a very important and pertinent problem in computational sciences. It is almost impossible to find exact solutions to many non-linear equations. For such problems, it would be important to develop methods to obtain approximate solutions. Therefore, we investigate an iterative technique for obtaining an approximate solution of the non linear equations of the form: where ϕ : D ⊂ R → R is substantially differential fucntion in the interval D. The well-known Newton's method to find the approximate solution of the non-linear equation (1) is given by , n = 0, 1, 2..... (2) It is one point second-order method and requires evaluation of one function and one derivative at each iteration. The one point iteration method based on the k-function evaluation has maximum order k [1,2]. The multipoint iterative approach has surpassed the theoretical limit of the onepoint methods in terms of computational efficiency and convergence order. If n be the number of function evaluations per iteration of the method and ρ represents the order of convergence, then the efficiency index [2] of the method can be measured by ρ 1 n . In the last few decades, the problems of finding an approximation to the root of non-linear equations have been extensively studied. Some surveys and complete literatures in this direction could be found in Argyros et al. [3], Abbasbandy [4], Chun [5,6], Kogan et al. [7], Kumar et al. [8], Thota [9], Mahesh et al. [10], Sabharwal [11], Sharma et al. [12], Chanu et at. [13] and the references therein. Some of the well-known methods developed recently for solving non-linear equations are detailed below: In 2007, Noor et al. [14], proposed following modified Housholder iterative method (NM1 for short) for solving nonlinear equations with order six: In 2007, Noor et al. [15], devised the following Predictor-Corrector Halley scheme for nonlinear equations (NM2 for short) with order six: In 2008, Parhi [16], presented a following new method for nonlinear equations (PM for short) with order six: In 2012, Chun et al. [17], suggested a following new scheme for nonlinear equation (CM for short) with order six: In 2014, Singh et al. [18], presented the following new method for nonlinear equations (SM for short) with order six: In 2019, Shengfen Li [19], proposed the following fourth order iterative method(LM for short) using Thiele's continued fraction: Inspired by the recent activities in this direction, we present a new sixth-order approach for solving non-linear equations by altering the Taylor's series expansion and composition techniques.

CONSTRUCTION OF THE PROPOSED METHOD WITH ANALYSIS OF CONVERGENCE
Let ϕ(x) be a differentiable real valued function defined on the interval D ⊂ R. Suppose that α ∈ D is a simple zero for non-linear equation ϕ(x) = 0 and let x n be a initial guess sufficiently close to α. By Taylor series quadratic approximation of ϕ(x) about the point x n , we get Assuming ϕ(x n+1 ) = 0 to obtain the next approximation(x n+1 ) of the root(α) of ϕ(x) in the above equation, we get Reordering above equation, we get Substituting the value of (x n+1 − x n ) from equation (8) on the right side of equation (11), we get Using Newton's method as the predictor and equation (12) as the corrector, we get the following method where The third order derivative evaluation is required to implement the method given in equation (13). We introduce an approximation of third derivative to overcome this drawback. Let y n = x n − ϕ(x n )/ϕ (x n ) and using the Taylor series about the point x n , we get Using y n − x n = −ϕ(x n )/ϕ (x n ) in above equation, we get Substituting the value of ϕ (y n ) from equation (17) into the equation (13), we get the following method free from third derivative evaluation of the function: where The implementation of the proposed method (NPM1) given by equation (18)  Hermite's interpolation. We consider that where q 1 , q 2 , q 3 and q 4 are unknowns that can be establish from the following conditions [20]: The above condition will create a four linear equations with four unknown variable q 1 , q 2 , q 3 & q 4 . The following expression can be obtained by solving those four linear equations [20]: Substituting the value of ϕ (y n ) from equation (20) into equation (18), we get the following iterative scheme free from the evaluation of higher derivatives: where M 1 = ϕ(x n ) 3 ϕ (y n ) 2 S(x n , y n ), M 2 = ϕ(x n ) 3 ϕ(y n )S(x n , y n ) 2 , M 3 = ϕ (x n )ϕ(y n )ϕ (y n ) and M 4 = ϕ(x n ) 2 S(x n , y n ) − 2ϕ(y n )ϕ (x n ) 2 .

NUMERICAL EXPERIMENTATION
In this section, we present the efficiency of the new proposed methods by applying the method on some nonlinear test function. The test functions and their initial guesses are listed in Table   1.
We compare the novel methods with the existing sixth order method given in (3), (4), (5), (6) and (7) denoted by NM1, NM2, PM, CM and SM respectively. We denote the methods given in equation (18) and (21) by NPM1 and NPM2 respectively. The results of numerical comparison on the test functions with their roots are summarized in Table 2 to Table 6. The absolute residual error (| ϕ(x n ) |) of the corresponding functions, the approximated root (x n ) and the total number of function evaluation (TNFE) after completion of four full iterations of methods is presented from Table 2 to Table 6. From the results available in Table 2 to Table 6, we conclude that the newly proposed method given by equation (21) provides a better estimation of roots than other existing methods.

CONCLUDING REMARKS
We have introduced a new sixth-order root-finding scheme for solving non-linear equations with four function evaluations. Taylor's series and composition technique are used to build the proposed scheme. The new iterative approach has an efficiency index of 1.56. Compared to other existing well known sixth-order schemes, numerical experimentation has shown that the newly proposed method is faster, uses fewer total number of function evaluations, and has a very low absolute residual error.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interest.