New Iterative Method with Application

In this paper, we consider iterative methods to find a simple root of a nonlinear equation f(x) = 0 , where f : D ∈ R → R for an open interval D is a scalar function.


Materials and Methods
Noor [6] proposed new fourth order method defined by Where ( ) / ( ). x f x f x  =− It is clear that to implement (3,2), one has to evaluate the second derivative of the function. This can create some problems. In order to overcome this drawback, several technique have been developed [2][3][4][5].
In [8],a second-derivative-free method is obtained through approximating the second derivative () n fy  in by In a recent paper, Noor and Khan [8] have used the same approximation of the second derivative (4) in (3) to suggest the following Iterative methods The following approximations of () n fy  are obtained in [10] α x n+1 where β∈ R. We then apply the approximations (6) and (7) to the method (5). Now, Combining (6) and (5), we get the new iterative method Using (7) in (5), we get a new family of iterative method However, for the following iteration scheme where and are parameters to be determined .
If we choose 2 and 1, we get the third order method.
which introduced by Chun in [11] If 0 and 1, then we obtain a third order method which was introduced by Xiaojian in [12].

Results and Discussion
All computations were done using the Mathematica package using 64 digit floating point arithmetic's.
We accept an approximate solution rather than the exact root, depending on the precision (ϵ) of the computer. We use the following stopping criteria for computer programs: | +1 − | ≺ and so, when the stopping criterion is satisfied, 1 n x + is taken as the exact root α computed. We used the fixed stopping criterion ϵ = 10 −15 .
We employ the present methods to solve some nonlinear equations, which not only illustrate the methods practically but also serve to check the validity of theoretical results we have derived. We present some numerical test results for various iterative schemes in Table 1.
We present some numerical test results for various iterative schemes in Table 2. Compared with the Newton method (NM), the method of Chun (13)(CM), Xiaojian (XM)(14). and the methods (14)(OM1) and (15) (OM2). The test results in Table 2 show that for most of the functions we tested, the methods introduced in the present presentation have at least equal performance compared to the other third-order method, and can also compete with Newtons method.

Conclusions
We have proposed two families of iterative methods for solving nonlinear equations. Numerical results show that the number of iterations of the new method are always less than that of the classical Newtons method and can be compared with other methods.