NEW ITERATIVE METHODS FOR SINGLE VARIABLE EQUATIONS

In this paper, we establish new iterative methods for the solution of scalar equations by using the decomposition technique due to Noor and Noor [Some iterative schemes for nonlinear equations, Appl. Math. Comput. 183 (2006), 774-779]. AMS Subject Classification: 65H05


Introduction
One of the most significant problems in numerical analysis is to find the solution of nonlinear equations.Various iterative methods have been studied to find the approximate roots of nonlinear equations, see [1,3,4,5,7,10,11,13] and references there in.These methods can be classified as one-step, two-step and three-step methods.In [5], Chun has proposed several one-step and two-step iterative methods with higher-order convergence by using the decomposition technique of Adomian [2].Many other iterative methods have also been developed for finding the simple roots of nonlinear equations.But there are many scientific problems in which the nature of the solution to the governing nonlinear equations is not simple.In order to find the multiple roots, Schröder [18] made the first contribution and introduced the following modified Newton's method: If m = 1, then the equation is the classical Newton's formula.
During the last two decades, much attention has been devoted by various researchers for solving nonlinear equation with multiple roots.Chun et al. [6], Chun and Neta [8], Homeier [12], Osada [17] have developed some techniques to find the multiple roots of nonlinear equations.In the recent years, the researchers have made significant and interesting contribution in this field [9,14,15].
In this paper, we propose some new iterative methods for finding multiple roots of scalar equations by using decomposition technique given in [16].
Consider the nonlinear equation It is well known that if α is a root with multiplicity m, then it is also a root of f ′ (x) = 0 with multiplicity m − 1 of f ′′ (x) = 0 with multiplicity m − 2 and so on.Hence if initial guess x 0 is sufficiently close to α, the expressions will have the same value.
Remark 1.1.The generalized Newton's formula gives a quadratic convergence when the equation f (x) = 0 has a pair of double roots in the neighbourhood of x 0 .It may be noted that for the double root α near to x 0 , f (α) = 0 = f ′ (α).

New Iterative Methods
We can rewrite the nonlinear equation (1.1) as a coupled system: where γ is the initial approximation for a zero of (1.1).We can rewrite (2.1) in the following form: where and Here N (x) is a nonlinear operator.
As in [16], the solution of (2.3) has the series form, The nonlinear operator N (x) can be decomposed as it has been shown in [16].Also the series ∞ i=0 x i converges absolutely and uniformly to a unique solution of equation (2.3) if the nonlinear operator x j . (2.8) Thus we have the following iterative scheme: x 2 = N (x 0 + x 1 ), . . . Then From (2.4), (2.5) and (2.9), we have and (2.12) It follows from (2.4), (2.9) and (2.10), that This enables us to suggest the following method for solving the nonlinear equation (1.1).
Algorithm 2.1.For the given x 0 compute the approximate solution x n+1 by the iterative schemes: which is known as the generalized Newton's formula and is quadratically convergent.

Convergence Analysis
The convergence analysis of Algorithms 2.2 and 2.3 is given in this section.Proof.Let α be a root of f (x) of multiplicity 2, then by expanding f (x n ) and f ′ (x n ), in Taylor's series about α, we obtain where e n = x n − α and c k = f (k) (α) k! , k = 2, 3, .... Using (3.1) and (3.2), we have Thus Using (3.5) and (3.8), we have

Numerical Examples
In this section, we present some numerical examples to demonstrate the performance of the newly developed iterative method.In the following Tables 1-12, we compare our proposed methods (Algorithms 2.2 and 2.3) (NIM1) and (NIM2) with classical Newton's method (NM), generalized Newton's method (GNM), Chun et al. [6,Equations (35) and (36] (BNM1) and (BNM2).All the computations for above mentioned methods are performed using software Maple and ε = 10 −10 as tolerance and also the following criteria is used for estimating the zero: Here N denotes the number of iterations.

Conclusions
In the present work, we have proposed two new iterative methods (NIM1) and (NIM2) with convergence order 2 for finding the multiple roots of nonlinear equations.The numerical results presented in the Tables 1-12 given in the previous section reveal that our iterative methods (NIM1) and (NIM2) are even comparable with the methods developed by Chun et al. [6] (BNM1) and (BNM2) with convergence order 3.The idea and technique employed in this paper can be developed to higher-order multi-step iterative methods for solving nonlinear equations having multiple roots.

Theorem 3 . 1 .
Assume that the function f : D ⊂ R → R for an open interval D has a multiple root α ∈ D of multiplicity 2. Let f (x) be sufficiently smooth in the neighborhood of the root α.Then the order of convergence of the methods defined by Algorithms 2.2 and 2.3 is 2.

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Expanding f (y n ) by Taylor's series about α, we get f (y n ) Now for Algorithm 2.3, we use (3.2) and (3.5) to compute z n as follows