An Efficient Family of Root-Finding Methods with Optimal Eighth-Order Convergence

We derive a family of eighth-order multipoint methods for the solution of nonlinear equations. In terms of computational cost, the family requires evaluations of only three functions and one first derivative per iteration. This implies that the efficiency index of the present methods is 1.682. Kung and Traub (1974) conjectured that multipoint iteration methods without memory based on n evaluations have optimal order . Thus, the family agrees with Kung-Traub conjecture for the case . Computational results demonstrate that the developed methods are efficient and robust as compared with many well-known methods. 1. Introduction Solving nonlinear equations is one of the most important problems in science and engineering [1, 2]. The boundary value problems arising in kinetic theory of gases, vibration analysis, design of electric circuits, and many applied fields are reduced to solving such equations. In the present era of advance computers, this problem has gained much importance than ever before. In this paper, we consider iterative methods to find a simple root of the nonlinear equation , where be the continuously differentiable real function. Newton’s method [1] is probably the most widely used algorithm for solving such equations, which starts with an initial approximation closer to the root and generates a sequence of successive iterates converging quadratically to the root. It is given by the following: In order to improve the local order of convergence, a number of ways are considered by many researchers, see [3–26] and references therein. In particular, King [3] developed a one-parameter family of fourth-order methods defined by where is the Newton point and is a constant. This family requires two evaluations of the function and one evaluation of first derivative per iteration. The famous Ostrowski’s method [4, 5] is a member of this family for the case . From practical point of view, the methods (1.2) are important because of higher efficiency than Newton’s method (1.1). Traub [5] has divided iterative methods into two classes, namely, one-point methods and multipoint methods. Each class is further divided into two subclasses, namely, one-point methods with and without memory, and multipoint methods with and without memory. The important aspects related to these classes of methods are order of convergence and computational efficiency. Order of convergence shows the speed with which a given sequence of iterates converges to the root while the computational efficiency concerns with the economy of the entire process. Investigation of one-point methods