SOLVING SYSTEM OF NONLINEAR EQUATIONS USING FAMILY OF JARRATT METHODS

: The aim of this paper is generalization fourth order Jarratt formula iterative methods for solving system of nonlinear equations (SNLE) of n-dimension with n-variables. We present three algorithms for solving (SNLE). We prove that these algorithms have convergence. Several numerical examples are tested of the new iterative methods. These new algorithms may be viewed as an extensions and generalizations of the existing methods for solving the system of nonlinear equations.

Wanga, et.al [9] Presented a variant of Jarratt method for solving non-linear equations and evaluated of the function at another point in the procedure iterated by Jarratt method, Sharma, et.al [10] presented a fourth order method for computing multiple roots of nonlinear equations.The method is based on Jarratt scheme for simple roots.Amat [11] used family of higher order iterative methods for solving nonlinear scalar equations, this family includes, as particular cases, some of the most famous third-order iterative methods: Chebyshev methods, Halley methods, super-Halley methods, C-methods and Newton-type two-step methods and analyzed the convergence of this family.Soleymani et.al [12] investigated the construction of some two-step without memory iterative classes of methods for finding simple roots of nonlinear scalar equations.
Khattri et.al [13] constructed a family of optimal fourth order iterative methods requiring three evaluations.Chun [14] proposed a new fourth order optimal root-finding methods for solving nonlinear equations and obtained the classical Jarratt's family of fourth-order methods are as special cases Maheshwari [15] solved nonlinear equations by using a fourth order numerical iterative method which has fourth order convergence.Geum et.al, [16] developed a new family of four-step optimal multipoint iterative methods of order sixteen for solving nonlinear equations.Soleymani, et.al [17] proposed a two-step class of without memory iterations for finding simple roots.Wang [18] modified Jarratt method with sixth-order convergence for solving non-linear equations.
Finally, Hafiz ([19], [20], [21]) modified some iterative schemes to get new algorithms for solving system of nonlinear equations.It is also remarked that methods in which there are two evaluations of the first-order derivative and one of evaluation of the function (with the fourth-order convergence) are called as optimal Jarratt-type methods in literature.For further reading on this topic, one may consult [22], [23].

Iterative Methods
Newton's iterative method is the best known method for finding a real or complex root x of the nonlinear equations f (x) = 0 which is given by [24] x This method converges quadratically in some neighborhood of x.It is also well-known that for any function H , with ∞, the iterative method can be written as [14]. Where Is iterative method of order 3. Now we obtain the optimal fourth-order methods for the Newton iteration function [14] can be written as. Where If we take H (t) = 1+ 1 2 ( t 1−t ) then eq (5), eq (6) lead the well-known Jarratt's forth order method [13], [14], [25] x Where If we take another H (t) = 1+ 9 6−4t − 9 6−2t in eq (5), eq (6) leads to another optimal fourth-order Jarratt's method [14], [25] x Where If we take another H(t) = 1+ t 2 + t 2 2 in in eq (5), eq (6) leads to another optimal fourth-order Jarratt's method [14], [25] x i+1 = x i − 1 + 3 4 Where Another optimal fourth-order Jarratt's method [14], [25] Recently, an efficient fourth-order technique in which we have two evaluations of the first derivative and one evaluation of the function had been presented by Khattri and Abbasbandy in [12] as comes next. where , ∝ 3 = 15 8 , ∝ 4 = 0, this formula can be written as [12]

System of Nonlinear Equations (SNLE)
The general form of a system of non-linear equations is 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 : 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 [24] 4. Modified Algorithms for Solving System of Nonlinear Equations Definition 1. Suppose that x be the simple zero of sufficiently differentiable functions and consider the numerical solution of the system of equations F (x) = 0, where F : D ⊆ R n → R n is a smooth mapping that has continuous second order partial derivatives on a convex open set D, and that has a locally unique root x in D, For solving system of nonlinear equations on n-dimensional and n-variables we develop fourth order iteration method in eq (7),equ (8) can be written as: Algorithm 1.For a given x 0 , compute the approximate solution x i+1 by iterative method schemes in eq (7),equ (8) we get: Algorithm 2. For a given x 0 , compute the approximate solution x i+1 by iterative scheme method of eq (9),equ (10) can be written as: Algorithm 3.For a given x 0 , compute the approximate solution x i+1 by iterative scheme in equation eq (14) can be written as:

Convergence Analysis
In this section, we consider the convergence of our algorithm using the Taylor's series technique.
Theorem 1.Let x * be a sample zero of sufficient differentiable function F :⊆ R n → R n for an open interval.If x 0 is sufficiently close to x * , then the two step method defined by our algorithm (3) has convergence is at least of order 3.

Proof. Consider to
where w n = 1 2 (x n + y n ), Let x * be a simple zero of F .Since F is sufficiently differentiable, by expanding F (x n ) and F ′ (x n ) about x * , we get and where 21) and ( 22), we have (see [26]) From ( 20) and ( 23), we get From ( 17), we get, and In general, according to [26]: Then we will have Finally, From ( 22) )which shows that Algorithm (1) is at least a third order convergent method, the required result.Since asymptotic convergence of Newton method is c 2 and from Theorem (1), we deduce that the convergence rate of our algorithm is better than the Newton's method.And the cubic convergent method is vastly superior to the linear and the quadratically convergent methods [27]
We present some examples to illustrate the efficiency of our proposed methods.Here, numerical results are performed by Maple 15 with 200 digits but only 14 digits are displayed.In Tables 1-5 we list the results obtained in Algorithms (1-3), which we called, Khirallah and Hafiz Methods (KHM1, KHM2, KHM3), respectively and compare them with Newton-Raphson method (NM), Hafiz and Bahgat method (HBM) [16], and Darvishi (DAM).The following stopping criteria is used for computer programs: and the computational order of convergence (COC) can be approximated using the following formula: Table 2 shows the number of iterations and the computational order of convergence (COC).||x (n+1) − x (n) || and the norm of the function F (x (n) ) is also shown in Table 2 for various methods.
Example 2. In a case two dimension, consider the following systems of nonlinear functions [28], Example 3. In a case three dimension, consider the following systems of nonlinear functions [28].In this subsection, we test HPM with some sparse systems with m unknown variables.In examples (4-6), we compare the NR method with the proposed HPM method focusing on iteration numbers [15].Example 4. Consider the following system of nonlinear equations: The exact solution of this system is X * = [0,0,...,0] T .To solve this system, we set x 0 = [0.5, 0.5, ..., 0.5] T as an initial value.Table 4 is shown the result.
Example 5. Consider the following system of nonlinear equations: One of the exact solutions of this system is X * = [1,1,...,1] T .To solve this system, we set x 0 = [2,2.,...,2] T as an initial value.The results are presented in Table 4.
whose exact solution is y = ln x.We consider the following partition of the interval: Let us define now If we discretize the problem by using the second order finite differences method defined by the numerical formulas then, we obtain a (m × 1) × (m × 1) system of nonlinear equations F 10 : We take X 0 with y as a starting point.In particular, we solve this problem for m = 50, 75 and 100.The numerical results for the above system of nonlinear equations are presented in Table 5.The number of iterations of methods HBM, KHM1, KHM2 and KHM3 are equal but method KHM2, . . .KHM3 has advantage of they are free from second derivatives, over methods NM and HBM, because the cost of computing second derivatives is very high, see Table 5.
In Table 1 -Table 5, we list the results obtained by modified iterations methods.As we see from this tables, it is clear that, in most cases, the result obtained by DAM, HBM, KHM1,. . .KHM3 are equivalent and they very superior to that obtained NM.

Conclusions
In this paper, we presented three new algorithms for solving the system of nonlinear equations by generalizing and applying fourth order Jarratt iterative methods and used these algorithms for the first time for solving initial value problem.These methods have the same efficiency as the other fourth-order methods in the literature.We conclude from the numerical examples that the proposed methods have at least equal performance as compared with the other methods of the same order.Moreover, our proposed methods provide highly accurate results in a less number of iterations as compared with Newton-Raphson method.

Table 1 :
3. Number of iterations for Example 1

Table 3 :
Number of iterations for Example 3 6.2.Large Systems of Nonlinear Equations

Table 5 :
Number of iterations for Examples 7