A New Fifth-Order Iterative Method for Finding MultipleRoots of Nonlinear Equations

In th is paper, we present a fifth-order method for find ing mult iple zeros of nonlinear equations. Per iteration, the new method requires two evaluations of functions and two of its first derivative. It is proved that the method has a convergence of order five. Finally, some numerical examples are g iven to show the performance of the presented method, and compared with some known methods.


Introduction
Solving nonlinear equations is one of the most important problems in numerical analysis. In this paper, we consider iterative methods to find a mu ltiple root α of mu ltip licity m, that is, α In recent years, some modifications of the Newton method for mult iple roots have been proposed and analysed [1,3,[5][6][7]10]. However, there are not many methods known to handle the case of mu ltip le roots. Hence we present a fifth-order method for finding mult iple zeros of a nonlinear equation and only use four evaluations of the function per iteration. In addition, the new methodhas a better efficiency index than the third and fourth order methods given in [1,3,4,7,10]. In v iew of th is fact, the new method is significantly better when compared with the established methods. Consequently, we have found that the new method is efficient and robust. The well-known Newton's method for finding mult iple roots is given by In this section we define a new fifth-order method. In order to establish the order of convergence of the new method we state the three essential definitions.

Definiti on 1 Let
where .

n ∈ 
The new fifth-order method for finding mu ltip le root of a nonlinear equation is expressed as Another form of the formu la (7) is given by x is the initial value and provided that the denominators of (6) and (7) are not equal to zero.
where n ∈  and Moreover by (6), we have By using (9) and (13), we get Since fro m (7) Substituting appropriate expressions in (18) and after simp lification we obtain the erro r equation The error equation (19) establishes the fifth-order convergence of the new method defined by (7).

The Established Methods
For the purpose of co mparison, we consider three fourth-order methods presented recently in [3,10] and the third-order method presented in [7]. Since these methods are well established, we shall state the essential expressions used in order to calculate the appro ximate solution of the given nonlinear equations and thus compare the effect iveness of the new fifth-ordermethod for mu ltip le roots.

Wu et al. Me thod
In [10], Wu et al. developed a fourth-order of convergence method, since this method is well established we state the essential expressions used in the method, where n ∈  , 0 x is the initial value and provided that the denominators of (21)-(22) are not equal to zero.

Li et al. Method 1
In [3], Li et al. developed some fourth-order convergence methods. The particular method we consider in this paper is expressed by, where ( ) where n ∈  , 0 x is the initial value and provided that the denominators of (23)-(25) are not equal to zero.

Li et al. Method 2
The second of the fourth-order of convergence method given in [3] is given as, where d is given by (29), n ∈  , 0 x is the initial value and provided that the denominators of (30)-(32) are not equal to zero.

Thukral Third-Or der Method
In [7], Thukral developed a third-order of convergence method; the particular expressions of the method are given as, x is the initial value and provided that the denominators of (36)-(37) are not equal to zero.

Application of the New Fifth-Order Iterative Method
The present fifth-order method given by (7) is employed to solve nonlinear equations with mu ltiple roots and co mpare with the Wu et al., two of Li et al. and Thukral methods, (22), (25), (30) and (37), respectively. To demonstrate the performance of the newfifth-order method, we use ten particular nonlinear equations. We determine the consistency and stability of results by examining the convergence of the new iterative methods. The findings are generalised by illustrating the effectiveness of the fifth-order methods for determining the mult iple root of a nonlinear equation. Consequently, we give estimates of the approximate solution produced by the methods considered and list the errors obtained by each of the methods. The numerical computations listed in the tables were performed on an algebraic system called Maple. In fact, the errors displayed are of absolute value and insignificant approximat ions by the various methodshave been omitted in the following tables.
The new fifth-order method requires four function evaluations and has the order of convergence five. To determine the efficiency index of the new method, we shall use the definition 2. Hence, the efficiency index of the fifth-order method given by (7) is 4 5 1.4953 ≈ whereas the efficiency index of the fourth-order and third-order methods is given as 4 Table 3.
3cos 5 x f x xe

Conclusions
In this paper, we have demonstrated the performance of a new fifth-order method for solving nonlinear equations with mu ltip le roots. Convergence analysis proves that the newmethod preserves its order of convergence.Furthermore, we have examined the effectiveness of the new fifth-order iterative method by showing the accuracy of the mu ltip le roots of several nonlinear equations. The main purpose of demonstrating the newmethod fordifferent types of nonlinear equations was purely to illustrate the accuracy of the approximate solution and the computational order of convergence. Finally, the advantages of the new iterative methods are; it is simple to co mpute, does not contain any long expressions of m, (see Li et al methods),has a better efficiency index than the methods considered and hence it may be considered a very good alternative to the classical methods.