Novel Computational Iterative Methods with Optimal Order for Nonlinear Equations

This paper contributes a very general class of two-point iterative methods without memory for solving nonlinear equations. The class of methods is developed using weight function approach. Per iteration, each method of the class includes two evaluations of the function and one of its first-order derivative. The analytical study of the main theorem is presented in detail to show the fourth order of convergence. Furthermore, it is discussed that many of the existing fourth-order methods without memory are members from this developed class. Finally, numerical examples are taken into account to manifest the accuracy of the derived methods. 1. Prerequisites One of the important and challenging problems in numerical analysis is to find the solution of nonlinear equations. In recent years, several numerical methods for finding roots of nonlinear equations have been developed by using several different techniques; see, for example, [1, 2]. We herein consider the nonlinear equations of the general form where is a real valued function on an open neighborhood and a simple root of (1.1). Many relationships in nature are inherently nonlinear, in which their effects are not in direct proportion to their cause. Accordingly, solving nonlinear scalar equations occurs frequently in scientific works. Many robust and efficient methods for solving such equations are brought forward by many authors; see [3–5] and the references therein. Note that Newton’s method for nonlinear equations is an important and fundamental one. In providing better iterations with better efficiency and order of convergence, a technique as follows is mostly used. The composition of two iterative methods of orders and , respectively, results in a method of order , [6]. Usually, new evaluations of the derivative or the nonlinear function are needed in order to increase the order of convergence. On the other hand, one well-known technique to bring generality is to use weight function correctly in which the order does not die down, but the error equation becomes general. In fact, this approach will be used in this paper. Definition 1.1. The efficiency of a method is measured by the concept of efficiency index, which is given by where is the convergence order of the method and is the whole number of evaluations per one computing process. Meanwhile, we should remember that by Kung-Traub conjecture [7] as comes next, an iterative multipoint scheme without memory for solving nonlinear equations has the optimal efficiency index and optimal rate of convergence . Higher-order methods are widely referenced in