A MODIFIED METHOD FOR SOLVING SYSTEM OF NONLINEAR EQUATIONS

Solving systems of nonlinear equation is a great im portant which arises in various branches of science  and engineering. In the last decades, several numer ical techniques were proposed to solve these problems. In this study, we propose a modified of i terative method which is based on the idea of Newton method and Fixed point iteration method. The proposed method has been illustrated with several examples from the reference. The numerical esults indicate that this proposed method provide the good performance of iterations.


INTRODUCTION
Solving systems of nonlinear equations is a great importance, because these systems frequently arise in various branches of pure and applied sciences.
The general form of a system of nonlinear equations is Equation 1: 1 1 2 n 2 1 2 n n 1 2 n f (x , x ,…, x ) = 0,f (x , x ,…, x ) = 0,f (x , x ,…, x ) = 0 (1) where, each function f i can be thought of as mapping a vector x = (x 1 , x 2 ,…,x n ) of the n-dimensional space R n , into the real line R. The system can alternatively be represented by defining a functional F, mapping R n into R n by. F(x 1 , x 2 ,…,x n ) = (f 1 (x 1 , x 2 ,…,x n ),…,f n (x 1 , x 2 ,…,x n )) T Using vector notation to represent the variables x 1 , x 2 ,…,x n , a system (1) can be written as the form: The functions f 1 , f 2 ,…,f n are called the coordinate functions of F (Burden and Farires, 2010).
Recently, several iterative methods have been used to solve nonlinear equations and the system of nonlinear equations (Awawdeh, 2009;Noor, 2010;Cordero et al., 2011;Sharma and Sharma, 2011;Vahidi et al., 2012). Wang (2011) using a third order family of Newton-Like iteration method for solving nonlinear equations; Ozel (2010) has considered a new decomposition method for solving the system of nonlinear equations. Saha (2010) has presented a modified method to solving nonlinear equations by hybridising the results of Newton method and fixed point iteration method. Kim et al. (2010) developed a new scheme for the construction of iterative methods for the solution of nonlinear equations and giving a new class of methods from any iterative method. Furthermore, several iterative methods have been developed for solving the system of nonlinear equations by using various techniques such as Newton's method, Revised Adomian decomposition method, homotopy perturbation method, Householder iterative method (Darvishi, 2009;Noor and Waseem, 2009;Hosseini and Kafash, 2010;Darvishi and Shin, 2011;Hafiz and Bahgat, 2012a;2012b;Noor et al., 2012).
It is the purpose of this study to introduce a new improvement of Newton method by fixed point iteration method. We extend the Saha (2010)

Description of an Iterative Method
Consider a nonlinear equation: We assume that the Equation (2) admits a unique solution x * .
In Newton method the all known iterative formula used to find the real root is Equation 3: In fixed point iteration method (2) will be rewritten in the form: Equation 4 which is equivalent to (2) will converge to a real root in the interval D if |g`(x)|<1 for all x in D provided the initial approximation x 0 is chosen in D.
Choose the initial approximation x 0 then (x 0 , g (x 0 )) is a point on the curve Equation 5: The equation of the tangent to the curve given by (5) at the point (x 0 , g (x 0 )) is Equation 6: Now we consider the line Equation 7: Substituting y = x in (6) we have: which produces the following iteration scheme Equations 8:

The N-Dimensional Case
The Newton method (Gautschi, 2011;Sauer, 2011) is commonly used for solving such systems Equation 9: where, F: Ω ⊆ R n →R n is defined Equation 10: where, F'(x k ) is the Jacobian matrix in point x k . In fixed point iteration method (9) will be rewritten in the form x = g(x) We rewrite Equation 8 to solve the nonlinear system F(x) = 0, this produces the following iteration scheme Equations 11: where, I is an identity matrix.

Numerical Examples
We present some examples to illustrate the efficiency of our proposed methods, we solve four systems of nonlinear equations and one of a nonlinear boundary value problem. The following tables show the Number of Iterations (NI) to receive the required solution. For all test problems the stop criteria is || F(x) || <10 −9 .

JMSS
this system, we set x 0 = (5, 4, 2) T as an initial value. The results are presented in Table 3.

2.CONCLUSION
In this study, we have demonstrated the applicability of the modified method for the system of nonlinear equations with the help of some concrete examples. The results show that: the proposed problem can be solved by the proposed method.