A Derivative-free Method of Eighth-order for Finding Simple Root of Nonlinear Equations

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we have constructed an optimal eighth-order method with four function evaluations to solve the non-linear equations. The proposed method is a three-step method in which no derivative is required. Our scheme is optimal in the sense of Kung and Traub. Moreover, some test functions have been also included to confirm the superiority of the proposed method. At the end, we have presented the basins of attraction of some existing methods along with our proposed method to illustrate their performances.


Introduction
In the proposed article, we have recognized the multi-point iterative method of eighth-order, to find the simple root α of the nonlinear equations f (x) = 0, where f : D ⊆ R → R is a scalar function on an open interval D and it is sufficiently differentiable in a neighborhood of α.It is well known that these methods, to solve nonlinear equations, have numerous applications in science and engineering.The proposed method is three-step derivative-free and optimal in the sense of Kung and Traub.Kung-Traub conjecture [1] states that the multi-point iterative methods, based on n evaluations could achieve optimal convergence order 2 n−1 and our proposed method is in concurrence with it, because it needs only four function evaluation per iteration i.e. n = 4.So as to achieve the new derivative-free method, we have swapped derivative with appropriate approximations based on divided difference.Relevant to the present study, a host of researchers also need a special mention at this juncture, for their work subsequent journals can be referred to [2]- [16].The subsequent substance of this article can be summarized as follows.In section 2, attention has been given to eighthorder derivative-free methods presented in literature, in order to compare the effectiveness of the proposed method, where all the derivative-free methods satisfy the Kung-Traub conjecture.Section 3, contains the main contribution of this article, where optimal derivative-free scheme is described and that the method has eighth-order convergence rate, under certain conditions on the weight functions has been proven, herewith.Afterwards in section 4, a numerical comparison between the proposed scheme and derivative-free scheme is provided.Demonstration on the basins of attraction for all the derivative-free methods on two test functions, using the programming package MAT HEMAT ICA 8 has been done.Finally, in section 5, the concluding remarks have been given for the readers.

Assortment from the Literature
In this section, let us review the following three-step derivative-free methods.In 2011, Thukral proposed an optimal eighth-order method, with two weight functions in the following form [3]: where n ∈ N, β ∈ R + , provided that the denominators in (2.1) are not equal to zero, which is denoted by RT1, for In 2012, a new optimal eighth-order derivative-free method was introduced by Thukral in [4], which uses four evaluations of function per iteration and is given by: which is denoted by RT2, for β = 1.Recently, Soleimani et al. in [5] has obtained the following eighth-order derivative-free method in 2013, which is an optimal method in the sense of Kung-Traub: where ) 2 , which is denoted by PM12, for β = 1 and γ = 1.Again, first author of the previous reference in 2013 has demonstrated two eighth-order methods without memory in [6], which are an optimal three-step methods in the sense of Kung-Traub: which is denoted by FS13, and which is denoted by FS16, where 3 Derivative-free scheme with optimal order of convergence In this ensuing section, we have derived a new derivative-free multi-step method of order eight.So as to set up the order of convergence of the proposed method, we have stated the important definitions.Definition 3.1.Let f (x) be a real function with a simple root α and let x n be a sequence of real numbers that converges towards α.The order of convergence m is given by where ζ is the asymptotic error constant and m ∈ R + .Definition 3.2.Let n be the number of function evaluations of the proposed method.The efficiency of the proposed method is measured by the concept of efficiency index ( [7], [8]) and defined as q 1 n , where q is the order of the method.Now let us consider the three-step cycle of [2], and generalize the novel scheme by using some suitable approximation of the derivative In the formula (3.6), we have three evaluations of function and one evaluation of derivative of first order.At this moment, the foremost confront is to estimate f ′ (x n ) as powerfully as probable, so that we can get a derivativefree scheme.To obtain this derivative-free scheme, we approximate the derivative by divided difference method.Consequently, the derivative in the first step of (3.6) is replaced by where . Now, substituting the value of (3.7) into (3.6) and introducing three weight functions in third step, we have where , without the index n.These weight functions G(ζ ), S(η) and N(ξ ), should be chosen such that the order reaches at local eight.It is going to be illustrated in the following theorem.

an initial approximation of α and G, S and N are sufficiently differentiable real functions satisfying the following conditions: G
Then the derivative-free method (3.8) have optimal order of convergence eight and satisfies the error equation stated below: where c j = f ( j) (α) j! for j = 1, 2, 3, ...

Proof.
Let e n = x n − α, be the error in the iterate x n .Now by applying the Taylor's series expansion and taking that f (α) = 0, we have and Again by Taylor's expansion, we have Now the first step of (3.8) gives Writing the Taylor's expansion for f (y n ) about α, we obtain Now from (3.10) and (3.16), we have Furthermore, we have Dividing (3.16) by (3.13), we get Using (3.20) in the second step of (3.8), we obtain Now, with the help of Taylor's theorem the expansion of f (z n ) about α is given below    (4) (1)| < ∞, the error equation in the most recent step will be attained as comes after that The convergence order of new derivative-free method defined by (3.8) is eight, which is established by the expression (3.24).
After selecting different types of weight functions satisfying the conditions mentioned in the above theorem, we can generate different classes of methods.We choose the weight functions as follows: by taking G ′′ (0) = 0, S ′′ (1) = 8 and N (4) (1) = 0. Then its error equation becomes And if its error equation is and by taking a 1 = 0, a 2 = 4 and a 3 = 0. Then its error equation becomes

Numerical examples
The following section covers the effectiveness of the new eighth-order derivative-free methods by employing it onto six numerical examples.We compare our proposed method for β = 1, denoted by OWD8 in equation (3.3) with the Thukral methods denoted by RT1(2.8) and RT2(2.7) in the equations (2.1) and (2.2) respectively, Soleimani et.al. method denoted by PM12(12) in equation ( 2.3) and Soleymani methods denoted by FS13(13) and FS16(16) in equation (2.4) and (2.5) respectively.Table 1 is furnished with the considered nonlinear test functions with their roots.The Numerical results are provided in Table 2 and 3. Table 2 and 3 contains the approximate solution generated by all the eighth-order derivative-free methods and we also mentioned the absolute error after third-iteration obtained by each method.From Table 2 and 3 it is clear that our proposed three-step derivative-free method give better feedbacks i.e. our method provide better results than the other methods in almost every case.Here the word NC stands for non convergence of the method.It can be observed from tables that the total number of iteration taken by our new method to evaluate α correct to 70 decimal places with total number of function evaluation are better or similar to the other eighth-order derivative-free methods.The computational order of convergence COC (see [17]), approximated using the formula to check the computational efficiency of the all derivative-free methods related to their theoretical rate of convergence.Next, we talk about the basins of attraction for a variety of derivative-free methods to demonstrate the effectiveness of the proposed method.We use the graphical tools of the efficient computer programming package MAT HEMAT ICA 8, to find the basins of attraction for complex functions of different methods.The figure has been created for the values of z 0 in the rectangle D = [−3, 3] × [−3, 3] ∈ C and we subsequently allot different colors based on the number of iteration to each complex point z 0 ∈ D according to the root at which the corresponding method starting from z 0 converges and we mark the point as white if the method does not converges.Now, we have considered two test functions (i) z 3 − 1 2z + 1, its zeros are {−1.16537,0.582687 + 0.720119i, 0.582687 − 0.720119i} and (ii) z 3 + 1 with the roots {−1., 0.5 + 0.866025i, 0.5 − 0.866025i}.The results for both these functions are described in Figure 1 and Figure 2, respectively.It is clear from the figures for both the test functions, that the best methods are OWD8 and RT1 and the rest of the methods RT2, PM12, FS13 and FS16 show so chaotic behaviors.The chaotic behavior normally lead to an unwanted zero in implementation and also makes the methods to be unreliable.It is also clear from the figure that the divergence area are very less in our case.
Table 1: Test functions and their roots.

Non-linear functions
Roots    International Scientific Publications and Consulting Services

Conclusion
In the present article, we have presented a three-step derivative-free method of optimal eighth-order convergence by means of divided difference and weight function technique.The efficiency index of the proposed method is 1.682 and the convergence analysis verifies that the new method has eighth order of convergence.For the comparison of the proposed derivative-free method with some of the existing derivative-free methods available in the literature, six numerical examples were considered, which established the effectiveness of our contributed method.We have also provided the dynamical aspects of some methods, to show that the modified eighth-order derivative-free method is better than some other derivative-free methods.

Table 2 :
Results of convergence for different derivative-free methods.

Table 3 :
Results of convergence for different derivative-free methods.