Generating Optimal Eighth Order Methods for Computing Multiple Roots

: There are a few optimal eighth order methods in literature for computing multiple zeros of a nonlinear function. Therefore, in this work our main focus is on developing a new family of optimal eighth order iterative methods for multiple zeros. The applicability of proposed methods is demonstrated on some real life and academic problems that illustrate the efﬁcient convergence behavior. It is shown that the newly developed schemes are able to compete with other methods in terms of numerical error, convergence and computational time. Stability is also demonstrated by means of a pictorial tool, namely, basins of attraction that have the fractal-like shapes along the borders through which basins are symmetric.


Introduction
Finding out the roots of nonlinear equations is an important task in numerical mathematics and has many advantages in engineering and applied sciences [1][2][3][4][5]. In the present work, we consider numerical methods for locating the multiple root α of multiplicity m of a nonlinear equation f (x) = 0. Mathematically, by multiple root α of multiplicity m, we mean that f (j) (α) = 0, j = 0, 1, 2, . . . , m − 1 and f (m) (α) = 0.
Based on the quadratically convergent modified Newton's method (see [6]) many higher order methods have been developed in literature. For example, see [7][8][9][10][11][12][13][14][15][16][17][18] and references shown therein. According to Traub's terminology (see [5]), the Newton's method (1) is a one-point optimal order method. In recent years, a number of two-point optimal fourth order methods have been proposed for multiple zeros (see [10][11][12][14][15][16][19][20][21][22][23][24][25]). Some non-optimal multipoint methods of sixth order are developed in [17,26]. More recently, multipoint methods with optimal eighth order convergence have also been proposed in the literature which are shown below: We are motivated by the research moving in the direction of developing optimal higher order methods. So, we attempt to propose a new class of iterative methods with optimal eighth order for computing multiple zeros. The methodology is based on weight function approach for the construction of a new scheme. Each member of the proposed scheme has optimal order in the sense of the classical Kung-Traub conjecture [5]. The efficiency and robustness of the proposed methods are demonstrated by performing several numerical problems. We observe that our methods have better results than those obtained by the existing methods.
The rest of the paper is summarized as follows. In Section 2, the scheme of eighth order method is developed and its convergence is studied. Some special cases of the family are explored in Section 3. Numerical experiments for several examples are performed in Section 4 to demonstrate the applicability and efficiency of the presented methods. Section 5 contains complex geometry based on the geometrical tool basins of attraction. Concluding remarks are given in Section 6.

Construction of the Method
This section contains the construction and convergence analysis of the proposed method with the main theorem. So, we consider the following three-step scheme whose first step is modified Newton iteration (1) for the known multiplicity m ≥ 1: where f (y n ) 1 m and the functions H : C → C and G : C → C are analytic in the neighborhood of '0'. In order to discuss the convergence analysis of iterative scheme, (8) the following theorem is proved: Theorem 1. Let f : C → C be an analytic function in a domain containing a multiple zero (say, α) having multiplicity m. Assume that starter x 0 is close enough to α, then the iteration scheme expressed by (8)  Proof. Let e n = x n − α be the error at n-th stage. Expanding f (x n ) about α by Taylor's expansion, we have that m! e m n 1 + C 1 e n + C 2 e 2 n + C 3 e 3 n + C 4 e 4 n + C 5 e 5 n + C 6 e 6 n + C 7 e 7 n + C 8 e 8 n + O(e 9 n ) (9) and where for k ∈ N. By using (9) and (10), we obtain that where ω i = ω i (m, C 1 , C 2 , . . . , C 8 ) are given in terms of m, C 1 , C 2 . . . C 8 with explicitly written two By using (9) and (12), we get expression of u as where η i = η i (m, C 1 , C 2 , . . . , C 8 ) are given in terms of m, C 1 , C 2 , . . . , C 8 with explicitly written one coefficient Inserting the expressions (9), (10) and (13) in the second step of scheme (8), we obtain that where P j = P j (m, C 1 , C 2 , . . . , C 8 ) and j = 1, 2, 3, 4. Expansion of f (z n ) about α leads us to the expression Using (9), (12) and (15), we get expressions of v and w where where Expansions of weight functions G(u) and H(v) in the neighborhood of origin '0' by Taylor series yield and Hence by substituting (9), (10), (13), (16), (18) and (19) into the last step of scheme (8), we obtain the error equation where Γ i = Γ i (m, H(0), C 1 , C 2 , . . . , C 7 ) are given in terms of m, H(0), C 1 , C 2 . . . C 7 with i = 1, 2, 3. From Equation (20), it is clear that we will obtain at least fifth order convergence when H(0) = 1. Using the value H(0) = 1 in Γ 1 = 0, we will obtain By inserting the expression H(0) = 1 and G(0) = 1 in Γ 2 = 0, we have Now, with the help of the above independent expressions H(0) = 1, G(0) = 1 and G (0) = 0 in Γ 3 = 0, we ascertain that Substituting H(0) = 1, (21), (22) and (23) in Equation (20), the error equation showing eighth order convergence is given by Thus, proof of theorem is established.

Some Special Cases of Weight Functions of G(u) and H(v)
We explore some special cases of our proposed method (8) by employing different forms of weight functions. In this regard, the following simple members of the proposed family are defined: Case 1. Let us describe the following weight functions satisfying the conditions established in Theorem 1: Then corresponding optimal eighth order iterative scheme is given by Case 2. Let us describe the following weight functions that satisfy the conditions of Theorem 2.1: Then, the eighth order iterative scheme is Case 3. Next, the following weight functions from the conditions of Theorem 2.1 are selected: For these functions, the corresponding eighth order method is

Numerical Results
This section is dedicated to test the efficiency and convergence of the proposed class. To do this, we consider the special cases of the proposed class, namely methods (25)- (27), denoted by NM1, NM2, and NM3, respectively. A total number of four test problems are selected for numerical testing. In addition, we want to compare our methods with other existing robust schemes of eighth order for multiple zeros given by Behl et al. [27,31] and Zafar et al. [28]. In this regard, we consider method (2) for a 1 = 1, a 2 = 1 by Behl et al. [27], two special cases of method (6) by Behl et al. [31] and two special cases method (7) (for A 2 = 1, P 0 = 1) by Zafar et al. [28], and denote them by BM1, BM2, BM3, FM1 and FM2, respectively.
Various problems considered for numerical testing are shown in Table 1. Calculations are performed in the Mathematica software using multiple-precision arithmetic. The computed numerical values shown in Tables 2-5 include: (i) number of iterations (n) that are required to find the solution with stopping criterion |x n+1 − x n | + | f (x n )| < 10 −350 (ii) values of the last three successive errors |x n+1 − x n |, (iii) residual error f (x n ), (iv) computational order of convergence (COC) and (v) elapsed time (CPU-time in seconds) in execution of a program, which is measured by the command "TimeUsed[ ]". The computational order of convergence (COC) is calculated by applying the formula (see [33])  [26]:  [18]:   From the above tables, we observe that the accuracy is increasing in the values of successive approximations, which points to the good convergence of the methods. The present methods also show consistent convergence behavior as compared to the existing methods. At the time when stopping criterion |x n+1 − x n | + | f (x n )| < 10 −350 is attained, the value ' 0 ' for |x n+1 − x n | is displayed. Computational order of convergence shown in the penultimate column of each table overwhelmingly supports the theoretical convergence of order eight. The CPU-time values in the last column of each table show that the new methods utilize less execution time than the time used by existing methods, which confirms the effectiveness of the proposed techniques. The main motive to apply these methods on different types of nonlinear equations is to illustrate the exactness of the obtained approximate solution and the convergence to the solution. Similar numerical tests, performed on a variety of numerical problems of different kinds, ensured the above remarks to a large extent.

Basins of Attraction
We aim to present the complex dynamical nature of new methods based on the geometrical tool, namely basins of attraction of the multiple zeros of a polynomial P(z). Study of basins of attraction provides an important information about the stability and convergence of numerical methods. Initially, this idea was floated by Vrscay and Gilbert [34]. In recent times, many authors have used this idea in their work, see, for example [35,36] and references given there. The basic definitions related to dynamical concepts of rational function associated with iterative methods can be found in [34].
To view the geometry in complex plane, we assess the attraction basins of the roots by applying the methods on some polynomials (see Table 6). The basins of attraction assessed are shown in Figures 1-3 for the considered polynomials. To plot basins we use rectangles R ∈ C of size [−2, 2] × [−2, 2] and [−3, 3] × [−3, 3], and assign different colors to the basins. Black color is assigned to the points for which the method is divergent.  From these graphics, one can easily check the behavior and stability of any iterative procedure. If an initial guess z 0 is chosen in a region where different basins meet each other, it is difficult to guess which zero is going to be attained by the method that starts in z 0 . So, the selection of z 0 in such a region is not preferable. Both the black zone (divergent zone) and the zones with different colors are not suitable to consider the initial guess z 0 when we want to acquire a unique root. The most adorable pictures can be seen along the boundaries between the basins of attraction. These boundaries have fractal-like pictures and belong to the cases where the method is more demanding with choice of initial point. At such regions, the dynamic behavior of the initial guess is more unpredictable.

Conclusions
In this research article, we have developed a class of optimal eighth order methods for locating multiple roots of nonlinear equations with known multiplicity. The analysis of the order of convergence has been discussed, that proves the order eighth under well-known assumptions regarding the nonlinear function whose zeros we are looking for. Some particular cases have been presented and their performance has been compared with well-known methods available in literature. The robustness of new algorithms can be judged by the fact that the accuracy in the successive approximations to the solution is much better compared to the accuracy of existing ones. Moreover, the used CPU time in execution of program is less than that of the CPU time taken by the existing techniques in majority of the cases. These conclusions have also been verified by similar numerical testing on many other different problems.