MODIFIED NEW SIXTH-ORDER FIXED POINT ITERATIVE METHODS FOR SOLVING NONLINEAR FUNCTIONAL EQUATIONS

In this paper, we present a modified new sixth-order fixed point iterative method for solving nonlinear functional equations and analyzed. The modified new sixth-order fixed point iterative method has convergence of order six and efficiency index 1.8171 which is larger than most of the existing methods and the methods discussed in Table 1. The modified new sixth-order fixed point iterative method converges faster than the methods discussed in Tables 1-6. The comparison tables demonstrate the faster convergence of the modified new sixth-order fixed point method. AMS Subject Classification: 65H05, 65D32


Introduction
The problem, to recall, is solving equations in one variable.We are given a function f and would like to find atleast one solution of the equation f (x) = 0. Note that, we do not put any restrictions on the function f ; we need to be able to evaluate the function; otherwise, we cannot even check that a given x = α is true, that is f (r) = 0.In reality, the mere ability to be able to evaluate the function does not suffice.We need to assume some kind of "good behavior".The more we assume,the more potential we have, on the one hand, to develop fast iteration scheme for finding the root.At the same time, the more we assume, the fewer the functions are going to satisfy our assumptions!This a fundamental paradigm in numerical analysis.
We know that one of the fundamental algorithm for solving nonlinear equations is so-called fixed point iteration method [15].
In the fixed-point iteration method for solving nonlinear equation f (x) = 0, the equation is usually rewritten as Considering the following iteration scheme and starting with a suitable initial approximation x 0 , we built up a sequence of approximations, say {x n }, for the solution of nonlinear equation, say α. the scheme will be converge to α, provided that (i) the initial approximation It is well known that the fixed point method has first order convergence.Kang et al. [18] described a new second order iterative method for solving nonlinear equations extracted from fixed point method by following the approach of [7] as follows: If g′(x) = 1, we can modify (1.1) by adding θ = −1 to both sides as: which implies that In order for g θ (x) to be efficient,we can choose θ such that g ′ θ (x) = 0, we yields θ = −g ′ (x), so that (1.3) takes the form For a given x 0 , we calculate the approximation solution x n+1 , by the iteration scheme This is so-called a new second order iterative method for solving nonlinear equations, which converges quadratically.
To prove (1.4) analytically, we use Taylor expansion, expanding (1.1) about the point x n such that If g ′ (x n ) = 0, we can evaluate the above expression as follows: If we choose x n+1 the root of equation, then we have This is the analytical prove of new second order iterative method [18] for solving nonlinear equations, which converges quadratically.From (1.5) one can evaluate This is so-called the modified new third-order iterative method [5] for non-linear equations, which converges cubically.
In this paper, a modified new sixth-order fixed point iterative method for solving nonlinear functional equations having convergence of order 6 and efficiency index 1.8171 extracted from the modified new third-order iterative method for solving nonlinear equations [5] motivated by the technique of Mc-Dougall and Wotherspoon [22] has been presented.The proposed modified new sixth-order fixed point iterative method applied to solve some problems in order to assess its validity and accuracy.

Main Results
Let f : X ⊂ R → R for an open interval X be a scalar function and consider that the nonlinear equation f (x) = 0 (or x = g(x)), where g(x) : X ⊂ R → R. Then we have modified new third-order iterative method for nonlinear equations By following the approach of McDougall and Wotherspoon [22], we develop a modified new sixth-order fixed point iterative method as follows: Initially, we choose two starting points x and x * .Then we set x * = x, our modified new sixth-order fixed point iterative method that we examine herein is given by which implies (for n ≥ 1) (2.3) where These are the main steps of our modified new sixth-order fixed point iterative method.
The value of x 2 is calculated from x 1 using g(x 1 ) and the values of first derivative of g(x) evaluated at 1  2 (x 1 + x * 1 ) (which is more appropriate value of the derivative to use than the one at x 1 ), and this same value of derivative is re-used in the next predictor step to obtain x * 3 .This re-use of the derivative means that the evaluations of the starred values of x in (2.3) essentially come for free, which then enables the more appropriate value of the derivatives to be used in the corrector step (2.4).

Convergence Analysis
Theorem 3.1.Let f : X ⊂ R → R for an open interval X and consider that the nonlinear equation f (x) = 0 (or x = g(x)) has a simple root α ∈ X, where g(x) : X ⊂ R → R be sufficiently smooth in the neighborhood of α.Then the convergence order of the modified new sixth-order fixed point iterative method given in (2.4) is at least six.
Proof.To analysis the convergence of the modified new sixth-order fixed point iterative method (2.4), let where Let α be a simple zero of f and f (α) = 0 (or g(α) = α).Then we can easily deduce by using the software Maple that Now, since H (vi) (α) = 0, then according to Theorem 1.1, the modified new sixth-order fixed point iterative method (2.4) has sixth order convergence.

Comparison of Efficiency Indices
Frontini and Sormani [11], Homeier [14] and Weerakoon and Fernando [30] have presented numerical methods having cubic convergence.In each iteration of these numerical methods three evaluations are required of either the function or its derivative.The best way of comparing these numerical methods is to express the of convergence per function or derivative evaluation, the so-called "efficiency" of the numerical method.On this basis, the Newton's method has an efficiency of 2 1 2 ≈ 1.4142, the cubic convergence methods have an efficiency of 3 1 3 ≈ 1.4422.Kuo [21] has developed several methods that each require two function evaluations and two derivative evaluations and these methods achieve an order of convergence of either five or six, so having efficiencies of 5 1 4 ≈ 1.4953 and 6 1 4 ≈ 1.5651, respectively.In these Kuo's methods the denominator is a linear combination of derivatives evaluated at different values of x, so that, when the starting value of x is not close to the root, this denominator may go to zero and the methods may not converge.Of the four 6-th order methods suggested in Kuo [21], if the ratio of function's derivatives at the two value of x differ by a factor of more than three, then the method gives an infinite change in x.That is, the derivatives at the predictor and corrector stages can both be the same sign, but if their magnitudes differ by more than a factor of three, the method does not converge.
Jarrat [16] developed a 4-th order method that requires only one function evaluation and two derivative evaluations, and similar 4-th order method have been described by Soleymani et al. [27].Jarrat's method is similar to those of Kuo's methods in that if the ratio of derivatives at the predictor and corrector steps exceeds a factor of three, the method gives an infinite change in x.Jarratt's methods is similar to those of Kou in that if the ratio of the derivatives at the predictor and corrector steps exceeds a factor of three, the method gives an infinite change in x.
While our modified new sixth-order fixed point iterative method (MNFIM) has an efficiency of 6

Applications
In this section we included some nonlinear functions to illustrate the efficiency of our developed modified two-step fixed point iterative method.We compare the modified new sixth-order fixed point iterative method with the fixed point method and the new second order iterative method [18] by choosing following test function, Comparison is shown in Tables 2-6.Table 2-6.Shows the numerical comparisons of the modified new sixthorder fixed point iterative method (MNFIM) with the fixed point method (FPM) and the new second order iterative method (NIM) [18].The columns represent the number of iterations N and the number of functions or derivatives evaluations N f required to meet the stopping criteria, and the magnitude |f (x)| of f (x) at the final estimate x n .

Conclusions
A modified new sixth-order iterative method (MNFIM) for solving nonlinear functions has been established.We can concluded from Tables 1-6 that 1.The efficiency index of MNFIM is 1.8171 which is larger than the efficiency index of most of the existing methods and the methods discussed in Table 1.
2. MNFIM has convergence of order six.3.By using some examples the performance of MNFIM is also discussed.MNFIM is performing very well in comparison to the fixed point method and the new iterative method (1.6) as discussed in Tables 2-6.

1 3
≈ 1.8171.The efficiencies of the methods we have discussed are summarized in Table1given below.

Table 1 .
Comparison of efficiencies of various methods

Table 4 .
Comparison of FPM, NIM and MNFIM

Table 5 .
Comparison of FPM, NIM and MNFIM

Table 6 .
Comparison of FPM, NIM and MNFIM