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
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