Abstract
Based on Ostrowski's method, a new family of eighth-order iterative methods for solving nonlinear equations by using weight function methods is presented. Per iteration the new methods require three evaluations of the function and one evaluation of its first derivative. Therefore, this family of methods has the efficiency index which equals 1.682. Kung and Traub conjectured that a multipoint iteration without memory based on evaluations could achieve optimal convergence order . Thus, we provide a new class which agrees with the conjecture of Kung-Traub for . Numerical comparisons are made to show the performance of the presented methods.
1. Introduction
In this paper, we consider iterative methods to find a simple root of a nonlinear equation , where is a scalar function on an open interval . This problem is a prototype for many nonlinear numerical problems. Newton’s method is the most widely used algorithm for dealing with such problems, and it is defined by which converges quadratically in some neighborhood of (see [1, 2]).
To improve the local order of convergence, many modified methods have been proposed in the open literature; see [3–17] and references therein. King [3] developed a one-parameter family of fourth-order methods, which is written as: where is a constant. In particular, the famous Ostrowski’s method [2] is a member of this family for the case , and it can be written as Kung and Traub [15] who conjectured that an iteration method without memory based on evaluations of or its derivatives could achieve optimal convergence order . Thus, the optimal order for a method with 3 functional evaluations per step would be 4. King’s method [3], Ostrowski’s method, and Jarrat’s method [16] are some of the optimal fourth-order methods, because they only perform three functional evaluations per step. Recently, based on Ostrowski’s or King’s methods, some higher-order multipoint methods have been proposed for solving nonlinear equations. Bi et al. developed a scheme of optimal order of convergence eight [17], estimating the first derivative of the function in the second and third steps and constructing a weight function as well in the following form: where is constant. Liu and Wang in [18] presented the following family of optimal order eight: where and are in . J. R. Sharma and R. Sharma in [12] produce optimal eighth-order method in the following form:
In this paper, based on Ostrowski’s method, we present a new family of optimal eighth-order methods by using the method of weight functions and we apply a few weight functions to construct families of iterative methods with high convergence and high efficiency index. The convergence analysis is provided to establish their eighth order of convergence. In terms of computational cost, they require the evaluations of only three functions and one first-order derivative per iteration. This gives 1.682 as efficiency index of the presented methods. The new methods are comparable with Bi et al.’s method, Liu and Wang’s method, and Sharma’s method. The efficacy of the methods is tested on a number of numerical examples.
We use the symbols , , and according to the following conventions [1]. If , we write or . If , we write or . If , where is a nonzero constant, we write or . Let be a function defined on an interval , where is the smallest interval containing distinct nodes . The divided difference with the th order is defined as follows: Moreover, we recall the definition of efficiency index (EI) as , where is the order of convergence and is the total number of function evaluations per iteration.
2. The Methods and Analysis of Convergence
In order to construct new methods, we consider an iteration scheme of the form This scheme includes three evaluations of the function and two evaluations of its first derivative. Therefore, this scheme has efficiency index equal to 1.516. To improve the efficiency index, we approximate by the divided difference [12] Now, we present a new family of optimal eighth-order Ostrowski-type iterative methods by using the method of weight functions as follows: where , and are three real-valued weight functions when without the index , should be chosen such that the order of convergence arrives at the optimal level eight. If is an approximation to the zero of , then the corresponding iterative method is defined by Regarding (10), let us define the errors We will use Taylor’s expansion about the zero to express , and as series in , and , respectively. Then, according to (11), we represent , and as Taylor’s polynomials in .
Assume that is sufficiently close to the zero of then and are close enough to . Let us represent real functions , and appearing in (10) by Taylor’s series about , Symbolic computations reported here to find candidates for , and have been carried out in a Mathematica environment. We will find the coefficients of the developments (14) using a simple program in Mathematica and an interactive approach explained by the comments C1–C5. First, let us introduce the following abbreviations used in this program (see Algorithm 1):
| ||||||||||||||||||||||||||||||||||||||||||||||||
: from the expression of the error , we observe that is of the form The iterative method will have the order of convergence equal to eight if we determine the coefficients of the developments appearing in (14) in such a way that , and (in (16)) vanish. We find these coefficients equalling shaded expressions in boxed formulas to 0. First, from Out[a4], we have Without loss of generality, we can take and hence, . In what follows, , , , and are calculated using the already found coefficients.
: from Out[a5], we see that the choice gives .
: we obtain choosing .
: vanishes if we choose simultaneously .
: substituting the quantities in the expression of , found in the described interactive procedure, we obtain where , .
According to the above analysis, we have proved the following theorem.
Theorem 1. Assume that . Suppose , and . If the initial point is sufficiently close to , then the sequence generated by any method of the family (10) has eighth order of convergence to if , and are any functions with
Remark 2. Any method of the family (10) uses four evaluations per iteration and has eighth-order convergence and satisfies the conditions (19), which accords with the conjecture of Kung-Traub [15] that a multipoint iteration without memory based on evaluations achieves optimal convergence order for the case .
3. The Concrete Iterative Methods
In what follows, we give some concrete iterative forms of scheme (10).
Method 1. The functions , and defined by satisfy the conditions of Theorem 1. A new family of two-parameter eighth-order methods is obtained and the error equation becomes
Method 2. The functions , and defined by satisfy the conditions of Theorem 1. A new family of two-parameter eighth-order methods is obtained and the error equation becomes
Method 3. The functions , and defined by satisfy the conditions of Theorem 1. A new family of two-parameter eighth-order methods is obtained and the error equation becomes The families (21), (24), and (27) achieve eighth-order convergence. Per iteration the presented methods require three evaluations of the function and one evaluation of its first derivative. If we assume that all the evaluations have the same cost as function one, we have that the family of methods (10) has the efficiency index of which is better than of Newton’s method, of King’s methods, of some methods with sixth-order convergence, and of some variants of Ostrowski’s method with seventh-order convergence [14].
4. Numerical Results
Now, Method 1 (21), Method 2 (24), and Method 3 (27) are employed to solve some nonlinear equations and compared with Bi et al.’s method (BM), (4), (with ), Liu and Wang’s method (LWM), (5), (with and ), and Sharma’s method (ShM), (6). The test functions of are listed in Table 1.
Numerical computations reported here have been carried out in a Mathematica 8.0 environment. Table 2 shows the difference of the root and the approximation to , where is the exact root computed with 800 significant digits () and is calculated by using the same total number of function evaluations (TNFE) for all methods. The absolute values of the function and the computational order of convergence (COC) are also shown in Table 2. Here, COC is defined by [19]
5. Conclusions
We have obtained a new family of variants of Ostrowski’s method. The convergence order of these methods is eight, which consist of three evaluations of the function and one evaluation of the first derivative per iteration, so they have an efficiency index equal to . Therefore, the family of methods agrees with the conjecture of Kung-Traub for . Numerical examples also show that the numerical results of our new methods, in equal iterations, improve the results of other existing three-step methods with eighth order convergence.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are grateful to the referees for their comments and suggestions that helped to improve the paper. This research was supported by Islamic Azad University, Hamedan Branch.