An Optimal Eighth-Order Family of Iterative Methods For Multiple Roots

In this paper, we propose a new family of efficient and optimal iterative methods for multiple roots with known multiplicity 1 ) (  m . We use weight function approach involving one and two parameters to develop the new family. An extensive convergence analysis is discussed in order to demonstrate the optimal eighth-order convergence of the proposed scheme. Finally, numerical and dynamical tests are presented which confirm the theoretical results established in this paper and illustrate that the proposed family is efficient among the domain of multiple root finding methods.


Introduction
The problem of solving nonlinear equation is recognized to be very old in history as many practical problems arising in nature are nonlinear.Various one-point and multi-point methods are presented to solve nonlinear equation or system of nonlinear equation [17,18,21].The above cited methods are designed for the simple root of nonlinear equations but the behavior of these methods are not similar when dealing with multiple roots of nonlinear equations.The well known Newton's method with quadratic convergence for simple roots of nonlinear equations decays to first order when dealing with multiple roots of nonlinear equations.These problems lead to minor troubles such as greater computational cost and severe troubles such as no convergence at all.The prior knowledge of multiplicity of roots make it easier to deal with these difficulties.The anomalous behavior of the iterative methods while dealing with multiple roots is well known at least since 19th century when Schröder [20] developed a modification of classical Newton's method to preserve its 2nd order of convergence for multiple roots.The nonlinear equations with multiple roots commonly arise from different topics such as complex variables, fractional diffusion or image processing, applications to economics and statistics(Lẻvy distributions) etc.By knowing the practical nature of multiple root finders, various one-point and multi-point root solvers have been developed in recent past [1, 3, 4, 6-9, 11-15, 19, 24] but most of them are not optimal as defined by Kung and Traub [10] which states that an optimal without memory method can achieve its convergence order at most n 2 requiring 1  n evaluations of functions or derivatives.According to Ostrowski [17], if O is the convergence order of an iterative method and n is the total number of functional evaluations per iterative step then the index is known as efficiency index of an iterative method.Sharma and Sharma [19] proposed the following optimal fourth order multiple root finder with known multiplicity m as follows: Geum et al. in [7], presented a non-optimal family of two-point sixth-order methods to find multiple roots given as follows: where, 1 ) ( and : ℂ 2 → ℂ is holomorphic in a neighborhood of (0,0) .
Following is a special case of their family: Another non-optimal family of three-point sixth-order methods for multiple roots by Geum et al. [8], is given as follows: where, The weight functions : ℂ → ℂ is analytic in a neighborhood of 0 and : ℂ 2 → ℂ is holomorphic in a neighborhood of (0,0) .Following is a special case of the family (1.4): The families (1.2) and (1.4) require four functional evaluations to produce sixth order convergence with the efficiency index 1.5650 = 6 4   1 and therefore are not optimal in the sense of Kung-Traub's conjecture [10].
Recently, Behl et al. [2] have proposed a family of optimal eighth order iterative methods for multiple roots given as: where, the weight functions : ℂ → ℂ and : ℂ 2 → ℂ are analytic functions in a neighborhoods of (0) and (0,0), respectively, with , ) ( a and 2 a are complex non-zero free parameters.We take particular case (27) for ( 1 of the family by Behl et al. [2] and denote it by BM as follows: , ) ( Most recently, second optimal eighth order scheme have been proposed by Zafar et al. [22], which is given as follows: where  1 ,  2 ∈ ℝare free parameters and the weight functions : ℂ → ℂ, : ℂ → ℂ and : ℂ → ℂ are analytic in the neighborhood of 0 with , ) ( From the eighth order family of Zafar et al. [22], we consider the following special case denoted by ZM : , ) ( Optimal iterative methods are more significant than the non-optimal ones, regarding their efficiency and convergence speed.Therefore, there was a need to develop optimal eighth-order schemes for finding multiple zeros ( 1 > m ) as well as simple zeros ( 1 = m ) because of their better efficiencies and order of convergence [17], in addition optimal schemes require a small number of iterations to obtain desired accuracy as compare to fourth and sixth-order methods of Sharma and Geum [7,8,19].In this paper, our main concern is to find the optimal iterative methods for multiple root  with known multiplicity  ∈ ℕ of a sufficiently differentiable nonlinear function :  ⊆ ℝ → ℝ where  is an open interval.We, in here, develop an optimal eighth order zero finder for multiple roots with known multiplicity 1  m .The beauty of the method lies in the fact that developed scheme is simple to implement with minimum possible number of functional evaluations.The family requires four functional evaluations to obtain eighth-order convergence with the efficiency index 1.6817 = 8 4   1 .
The rest of the paper is organized as follows: In Section 2, we propose a new family of optimal eighth-order iterative methods to find multiple roots of nonlinear equations and discuss its convergence analysis.Some special cases are given in Section 3. In Section 4, numerical performance and comparison of the proposed schemes with the existing ones are given, dynamical analysis is given in section 5.Concluding remarks are given in Section 6.

Development of the scheme
In this section, we propose a new family of eighth-order method for a known multiplicity 1  m of the desired multiple root as follows: and the weight function : ℂ → ℂ is analytic function in the neighborhood of 0 and weight function : ℂ 2 → ℂ is holomorphic in the neighborhood of (0,0) and s t, and u are m to one   multiple -valued functions.
In the next theorem, it is demonstrated that the proposed scheme (2.1) achieves the optimal eighth order of convergence without increasing the number of functional evaluations.
(say) be a multiple zero with multiplicity 1  m of an analytic function Then the family of iterative methods defined by (2.1) has eighth-order convergence when the following conditions are satisfied: 2) Then the proposed scheme (2.1) satisfies the following error equations: where by the Taylor's series expansion (with the help of computer algebra software Mathematica), we obtain  ,  where   =   (,  1 ,  2 , … ,  8 ) are expressed in terms of ,  1 ,  2 , … ,  8 where the two coefficients 0 G and 1 G can be explicitly written as With the help of Taylor's series expansion, we obtain

Special Cases of Weight Functions
From Theorem 1, several choices of weight functions can be obtained, we have considered the following: Case 1: The polynmial form of the weight function satisfying conditions (2.2) can be represented as: A particular iterative method related to (3.1) is given by: SM-1: Case 2: The second suggested form of the weight functions in which ) (t k f is constructed using rational weight function satisfying conditions (2.2) is given by: The corresponding iterative method (3.3)can be presented as: (3.5)The corresponding iterative method obtained using (3.5) is given by: SM-3:

Numerical tests
In this section, we show the performance of the presented iterative family (2.1) by carrying out some numerical tests and comparing the results with existing method for multiple roots.All the numerical computations have been performed in Maple 16 programming package using 1000 significant digits of minimum number of precision.In that case  is not exact, it is replaced by a more accurate value which has more number of significant digits than the assigned precision.The test functions along with their roots  and multiplicity m are listed in Table 1 [16].The proposed methods SM-1 (3.2), SM-2 (3.4) and SM-3(3.6)are compared with the methods of Geum et al. given in (1.3) and (1.5) denoted by GKM-1 and GKM-2 and with method of Bhel given in (1.7) denoted by BM and Zafar et.al method given in (1.9) denoted by ZM respectively.Tables 2-8 display the errors of approximations to the sought zeros ( ) produced by different methods at the first three iterations, where . The initial approximation 0 x for each test function and computational order of convergence (COC) is also included in these tables, which is computed by the following expression [23]: .) )/( ( log It is observed that, the performance of new method SM-2 is same as method of BM for function 1 f and better than method of ZM for function 2 f .The newly developed schemes SM-1, SM-2 and SM-3 are not only convergent but also their speed of convergence is better than methods of GKM-1 and GKM-2.On the other hand methods of ZM and BM show divergence for function 3 f .For 4 f , 5 f , 6 f and 7 f the newly developed schemes newly developed schemes SM-1, SM-2 and SM-3 are comparable with methods of ZM and BM.Hence, we conclude that the proposed family is comparable and robust among existing methods for multiple roots.

Table 1: Test functions Test Functions
Exact root 

Dynamical analysis
Regarding the stability comparison, we use the routines presented in [5] for plotting the dynamical planes corresponding to each method (SM-1,SM-2, S-M3, BM and ZM) for the non-linear functions . For this, we define a mesh of 400  400 points, as each point of the mesh is an initial guess for the analyzed method on the specific nonlinear function.If the sequence of iteration method reaches (closer than 3 10  ) the multiple root in less than 80 iterations, then this point is painted in orange color; if the iterate converges to another thing (strange fixed points, cycles, etc.) then the point is painted black.The multiple root is represented in the different figures by a white star.
We observe from Figures 1-7 that the only basin of attraction is that of the multiple root (that is, the set of initial points converging to it fills all the plotted region of the complex plane), plotted in orange in the figures; although in general, convergence to other roots, divergence or even convergence to other fixed points that are not roots of the non-linear function (known as strange fixed points), can appear.We see in the figures, that the orange region is more bigger and brighter for the proposed schemes SM1, SM2 and SM3 than the regions of methods BM and ZM for all examples, that confirms their stability and fast convergence speed.
respectively.By using the above expressions (2.4) and (2.5) in the first substep of (2.1), we obtain )

. 4 ) 3 :
CaseThe third suggested form of the weight function in which )

1 : 2 : 3 : 4 : 5 : 6 : 7 :
Basins of attraction of different methods for f1 Basins of attraction of different methods for f2 Basins of attraction of different methods for f3 Basins of attraction of different methods for f4 Basins of attraction of different methods for f5 Basins of attraction of different methods for f6 Basins of attraction of different methods for f7

Table 2 :
Comparison of different methods for multiple roots

Table 3 :
Comparison of different methods for multiple roots

Table 4 :
Comparison of different methods for multiple roots

Table 5 :
Comparison of different methods for multiple roots

Table 6 :
Comparison of different methods for multiple roots

Table 7 :
Comparison of different methods for multiple roots

Table 8 :
Comparison of different methods for multiple roots