A New Third-Order Derivative Free method for Solving Nonlinear Equations

In the present paper, by approximating the derivatives in the Newton-Steffensen third-order method by central difference quotient, we obtain a new modification of this method free from derivatives. We prove that the method obtained preserves their order of convergence, without calculating any derivative. Finally, numerical tests confirm that our method give the better performance as compare to the other well known derivative free Steffensen type methods.


Introduction
A large number of papers have been written about iterative methods for the solution of the nonlinear equations. In this paper, we consider the problem of finding a simple root x * of a function f : D ⊆ ℜ −→ ℜ, i.e. f (x * ) = 0 and f ′ (x * ) ̸ = 0. The famous Newton's method for finding x * uses the iterative method starting from some initial value x 0 . The Newton's method is an important and basic method which converges quadratically in some neighborhood of simple root x * . However, when the first order derivative of the function f (x) is unavailable or is expensive to compute, the Newtons method is still restricted in practical applications. In order to avoid computing the first order derivative, Steffensen in [1] proposed the following derivative-free method. It is well known that the forward-difference approximation for the Newton's method becomes which is the famous Steffensen's method [1]. The Steffensen's method is based on forward-difference approximation to derivative. This method is a tough competitor of Newton's method. Both the methods are quadratic convergence, both require two functions evaluation per iteration but Steffensen's method is derivative free. The idea of removing derivatives from the iteration process is very significant. Recently, many high order derivative-free methods are built according to the Steffensen's method, see [ [2], [3], [4], [5], [6], [7], [8], [9]] and the references therein.
Again Dehghan et al. [9] introduced a new third-order Steffensen type method (Dehghan Method III): Recently Cordero et al. [2] presented a fourth-order Steffensen type method (Cordero Method): Other Steffensen type methods and their applications are discussed in [ [3], [4], [5], [6]]. The purpose of this paper is to develop a new third-order derivative-free method and give the convergence analysis. This paper is organized as follows. In Section 2, we present a new two-step third-order iterative method for solving nonlinear equations. In this method we approximate the derivative of the function by central difference quotient. The new method is free from derivative. We prove that the order of convergence of the new method is three. Numerical examples show better performance of our method in section 4. Section 5 is a short conclusion.

Development of the method and analysis of convergence
Let us consider the third-order Newton-Steffensen method [10]: . (2.1) Approximating the derivative f ′ (x n ) in (2.1) by the central-difference we obtain a new method free from derivatives, that we call the modified Newton-Steffensen method free from derivative (MNSDF): Now we are going to prove the method MNSDF has order of convergence three. and Further more it can be easily find

Numerical Tests
In this section, in order to compare the our new method with Steffensen method, Jain method, Dehghan method I, Dehghan method II, Dehghan method III and Cordero method, we give some numerical examples. For this consider the following functions:

Non-linear functions
Roots 0.767653 Table 2-6 shows the comparison of these methods for these functions. All the numerical computations have been carried out using MATHEMATICA 8. The numerical results show that the our proposed method is efficient. Table 2. Errors Occurring in the estimates of the root of function f 1 (x) = sin 2 (x) − x 2 + 1 after third iteration by the method described with initial guess x 0 = 1.
Methods Table 3. Errors Occurring in the estimates of the root of function f 2 (x) = x 2 − e x − 3x + 2 after third iteration by the method described with initial guess x 0 = 0.7.   Table 5. Errors Occurring in the estimates of the root of function f 4 (x) = cos(x) − xe x + x 2 after third iteration by the method described with initial guess x 0 = 1.  Table 6. Errors Occurring in the estimates of the root of function f 5 (x) = e x − 1.5 − arctan(x) after third iteration by the method described with initial guess x 0 = 1.