THREE-STEP ITERATIVE METHOD WITH EIGHTEENTH ORDER CONVERGENCE FOR SOLVING NONLINEAR EQUATIONS

In this paper, we propose and discuss a new higher-order iterative method for solving nonlinear equations. This method based on a Halley and Householder iterative method and using predictor-corrector technique. The convergence analysis of our method is discussed. It is established that the new method has convergence order eighteen. Numerical tests show that the new method is comparable with the well-known existing methods and gives better results. AMS Subject Classification: 41A25, 65H05, 65K05


Introduction
Finding iterative method for solving nonlinear equations is an important area of research in numerical analysis at it has interesting applications in several branches of pure and applied science can be studied in the general framework of the nonlinear equations f (x) = 0 (see [2]).Due to their importance, several numerical methods have been suggested and analyzed under certain condition.These numerical methods have been constructed using different techniques such as Taylor series, homotopy perturbation method and its variant forms, quadrature formula (see [3]), variational iteration method, and decomposition method.For more details, see [1]- [14].In this paper,based on a Halley and Householder iterative method and using predictor-corrector technique, we construct modification of Newton's method with higher-order convergence for solving nonlinear equations.The error equations are given theoretically to show that the proposed technique has eighteenth -order convergence.Commonly in the literature the efficiency of an iterative method is measured by the efficiency index defined as I ≈ p 1/d (see [19]), where pis the order of convergence and dis the total number of functional evaluations per step.Therefore this method has efficiency index 18 1/8 ≈ 1.435 which are higher than 2 1/2 ≈ 1.4142 of the Steffensen's method (SM) (see [4], [16], 3 1/4 ≈ 1.3161 of the (DHM) method [17].Several examples are given to illustrate the efficiency and performance of this method.

Iterative Method and Convergence Analysis
Consider the nonlinear equation of the type For simplicity, we assume that α is a simply root of (2.1) and γ is an initial approximation, sufficiently close to α.Using the Taylor's series expansion of the function f (x), we have .
This equality allow us to suggest the following iterative method for solving the nonlinear equation (2.1).Algorithm 2.1.For a given x 0 , find the approximate solution x n+1 by the iterative scheme It is well known that Algorithm 2.1 has a quadratic convergence.
Algorithm 2.2.For a given x 0 , compute approximates solution x n+1 by the iterative scheme This is known as Halley's method has cubic convergence (see [1,8]).
Algorithm 2.3.For a given x 0 , compute approximates solution x n+1 by the iterative schemes This is known as Householder's method and has cubic convergence, see [18].Now we suggest and analyze a new three-step iterative method for solving the nonlinear equation.If we using classical Newton-Raphson method Algorithm 2.1 as a predictor and Algorithm 2.2 and Algorithm 2.3 as a corrector, which is the main motivation of this paper.
Algorithm 2.4.For a given x 0 , compute approximates solution x n+1 by the iterative schemes Algorithm 2.4 is called the predictor-corrector Newton-Halley method (PCNH) and has eighteen-order convergence.Let us now discuss the convergence analysis of Algorithm 2. 4.
Theorem 2.1.Let r be a sample zero of sufficient differentiable function f :⊆ R → R for an open interval I.If x 0 is sufficiently close to r, then the three-step iterative method defined by Algorithm 2.4 has eighteenth-order convergence.
Proof.Let rbe a simple zero of f .Since f is sufficient differentiable, by expanding f (x n )and f ′ (x n )about r, we get where f ′ (r) , k = 2, 3, . .., and e n = x n − r.From (2.5) and (2.6), we have Let us set W = w n − r.Then the equation (2.7) can be re-written in the form Now expanding f (w n ), f ′ (w n ), f ′′ (w n ) about r and using (2.7), we have ) Combining (2.7)-(2.11),we have (2.12) Also expanding f (y n ), f ′ (y n ), f ′′ (y n ) in r and using (2.12), we have Combining (2.12)-(2.15),we obtain (2.16) Finally, wehave which shows that Algorithm 2.4 has eighteenth-order convergence.
Remark.The order of convergence of the iterative method 2.4 is 18.Per iteration of the iterative method 2.4 requires three evaluations of the function, three evaluation of first derivative, and two evaluation of second derivative.We take into account the definition of efficiency index (see [4], [15]), if we suppose that all theevaluations have the same cost as function one, we have that the efficiency index of the method 2.4 is 18 1/8 ≈ 1.435, which is better 2 1/2 ≈ 1.4142 of the Steffensen's method (SM) (see [16]), 3 1/4 ≈ 1.3161 of the DHM method, see [17].

Numerical Examples
For comparisons, we have used the fourth-order Jarratt method [19] (JM) and Ostrowski's method (OM) [15] defined respectively by and We consider here some numerical examples to demonstrate the performance of the new modified iterative method, namely (PCNH).We compare the classical Newton's method (NM), Jarratt method (JM), the Ostrowski's method (OM) and(PCNH).In the Table 1 and Table 2 the number of iteration is n = 3 for all our examples.But in Table 1 our examples are tested with precision ε = 10 −1000 .The following stopping criteriais used for computer programs: The computational order of convergence (COC ) can beapproximated using the following formula Table 1 shows the difference of the root r and the approximation x n to r, where r is the exact rootcomputed with 2000 significant digits, but only 25 digits are displayed for x n .In Table 2, we listed the number of iterations for various methods.The absolute values of the function f (x n ) and the computational order of convergence (COC ) are also shown in Tables 2, 3.All the computations are performed using Maple, Version 15.The following examples are used for numerical testing: Results are summarized in Tables 1, 2, 3 as it shows new algorithms are comparable with all of the methods and in most cases give better or equalresults.Table 2 Comparison of different methods.

Conclusions
In this paper, we have suggested and analyzed new higher-order iterative method and used for solving of nonlinear equations.This method based on a Halley and Householder iterative method and using predictor-corrector technique.The error equations are given theoretically to show that the proposed technique haseighteenth-order convergence.The new method attain efficiency index of 1.435, which makes it competitive.In addition, the proposed method has been tested on a series of examples published in the literature and show good results when compared it with the previous literature.

Table 2 :
Comparison of different methods

Table 3 :
Comparison of different methods