New Ninth Order J-Halley Method for Solving Nonlinear Equations

In the paper [1], authors have suggested and analyzed a predictor-corrector Halley method for solving nonlinear equations. In this paper, we modified this method by using the finite difference scheme, which had a quantic convergence. We have compared this modified Halley method with some other iterative methods of ninth order, which shows that this new proposed method is a robust one. Some examples are given to illustrate the efficiency and the performance of this new method.


Introduction
In recent years, several iterative type methods have been developed by using the Taylor series, decomposition and quadrature formulae (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and the references therein).Using the technique of updating the solution and Taylor series expansion, Noor and Noor [1] have suggested and analyzed a sixth-order predictor-corrector iterative type Halley method for solving the nonlinear equations.Also, Kou et al. [2][3][4] have also suggested a class of fifth-order iterative methods.In the implementation of these methods, one has to evaluate the second derivative of the function, which is a serious drawback of these methods.To overcome these drawbacks, we modify the predictor-corrector Halley method by replacing the second derivatives of the function by its finite difference scheme.We prove that the new modified predictor-corrector method is of fifth-order convergence.We also present the comparison of the new method with the methods of Kou et al. [2][3][4] and Hu et al. [5].In passing, we would like to point out that the results presented by Kou et al. [2][3][4] are incorrect.We also rectify this error.
Several examples are given to illustrate the efficiency and robustness of the new proposed method.

Iterative Methods
The Jarratt's fourth-order method [6] which improves the order of convergence is defined by Algorithm 1 where Recently, Kou et al. [2] considered the following twostep iteration scheme Algorithm 2 We now state some fifth-order iterative methods which have been suggested by Noor and Noor [6] and Kou et al. [2,3] using quite different techniques.
which is a two-step Halley method of fifth-order convergent.
In a recent paper Kou et al. [2,3] have suggested following iterative methods.
Algorithm 4 (SHM [3]).For a given x 0 , compute the approximate solution xnþ1 by the iterative schemes: where 0 , Algorithm 5 (ISHM [2]).For a given x 0 , compute the approximate solution xnþ1 by the iterative schemes: where 0 . 2 On the basis of the above discussion a new iterative technique is proposed below (named as FAJH): where

Analysis of Convergence
In this section, we compute the convergence order of the proposed method (FAJH).

Theorem: Let I   be a simple zero of sufficiently differentiable function for an open interval
x is close to ,  then the three-step algorithm 6 has ninth order of convergence.
Proof: The iterative technique is given by where Let  be a simple zero of f .By Taylor's expansion, we have, where .
Using , and we have (3.4) From , and we have (3.5)3.7 Thus we observe that the new three-step method (FAJH) has ninth order convergence.

Numerical Examples
In this section now we consider some numerical examples (see Table 1) to demonstrate the performance of the newly developed iterative method.We compare classical Newton method (NW), Kou et al. method (see, [2]) (VCM) and (VSHM), Noor et al. methods (see [1]) (NR1), (NR2) and also ninth order Zhongyong Hu et al. (Z Hu) [5] with the new developed method (FAJH).All the computations for above mentioned methods, are performed using software Maple , precision digits and as tolerance and also the following criteria is used for estimating the zero: 1) 3) Maximum numbers of iterations 500. We used the following examples for comparison:

Conclusion
In Tables 2-11, we observe that our iterative method      (FAJH) is comparable with all the methods cited in the above mentioned tables and gives better results even than ninth orders method of Hu et al. [5].With the help of the technique and idea of this paper, one can develop higherorder multi-step iterative methods for solving nonlinear equations, as well as a system of nonlinear equations.