A New Three Step Iterative Method without Second Derivative for Solving Nonlinear Equations

In this paper , an efficient new procedure is proposed to modify third –order iterative method obtained by Rostom and Fuad [Saeed. R. K. and Khthr. F.W. New third –order iterative method for solving nonlinear equations. J. Appl. Sci .7(2011): 916-921] , using three steps based on Newton equation , finite difference method and linear interpolation. Analysis of convergence is given to show the efficiency and the performance of the new method for solving nonlinear equations. The efficiency of the new method is demonstrated by numerical examples.


Introduction:
Iterative methods play a crucial role in approximating the solution of nonlinear equation ( ) . One of the most famous iterative methods is the classical Newton method, which is a well known basic method and possesses quadratic order of convergence. Several different methods were developed by number of researchers for the computation the root of the nonlinear equations , these numerical methods have been suggested using Taylor series , Homotopy analysis method ,Hermit interpolation method and other techniques , see [1][2][3][4][5][6][7][8][9][10][11] .
In this paper , a new three-step iterative method was suggested without second derivative by considering a suitable approximate of ( ) [12] using Taylor series to find a simple root of nonlinear equation. The new method is sixth-order convergent was proved.
Several examples are presented showing the accuracy of the proposed method .

Construction of new method and convergence criteria:
To illustrate the procedure of constructing our new iterative method , the following iterative method was considered: Algorithm 1.1: For a given , compute the approximate solution by the iterative scheme: which is known as Newton method and has quadratic convergence . Rostom and Fuad [13] have obtained the following third order iterative method.

Algorithm 1.2:
For a given , compute the approximate solution by the iterative scheme: (2) Substituting (1) into (Algorithm 1.2) , then the following iterative scheme was constructed: Algorithm 1.3: For a given , compute the approximate solution by the iterative scheme:

Open Access
To obtain a new three-step method of order six , a Newton method was used to rise the order of convergence , we have

Algorithm 1.4:
For a given , compute the approximate solution by the iterative scheme: Using the linear interpolation on two points ( ( ) ) and ( to obtain the approximation Combining (5) and (8) , the following new iterative method free from second derivative was suggested for solving nonlinear equation.
Algorithm 1.5: For a given , compute the approximate solution by the iterative scheme: For the new method defined by (Algorithm 1.5), We have the following analysis of convergence .
Theorem 1: let be the scalar function sufficiently smooth in the real open domain D, and r is a simple zero of .If is sufficiently close to r , then the method defined by (Algorithm 1.5) has six-order convergence , satisfied the following error equation According to (11), (12) 22) ,we conclude that the new iterative method defined by (9) in (Algorithm 1.5) has a sixth-order convergence.

2-Numerical implementations and Discussion:
In this section, some test problems are presented to compare the efficiency of the proposed method (TSM) . The following table (1) given the test functions and their solution r , found up to the 10 th decimal places.

Discussion:
In this paper, a new three -step iterative method without second derivative was suggested for finding roots of nonlinear equations. Analysis of the convergence is performed to show that the convergence order of the new method is six. Numerical results indicate that the new method takes a less number of iteration than the other methods.