Abstract
In this paper we construct a new third-order iterative method for solving nonlinear equations for simple roots by using inverse function theorem. After that a class of optimal fourth-order methods by using one function and two first derivative evaluations per full cycle is given which is obtained by improving the existing third-order method with help of weight function. Some physical examples are given to illustrate the efficiency and performance of our methods.
Highlights
Nonlinear equations plays an important role in science and engineering
We prove that order of convergence of this method is three
The following theorem indicates under what conditions on the weight functions and constant a in (3.2), the order of convergence will arrive at the optimal level four: Theorem 3.1
Summary
Nonlinear equations plays an important role in science and engineering. Finding an analytic solution is not always possible. The classical Newton’s method is the best known iterative method for solving nonlinear equations. Let β be the number of function evaluations of the new method. Kung and Traub [10] presented a hypothesis on the optimality of roots by giving 2n−1 as the optimal order. This means that the Newton iteration by two evaluations per iterations is optimal with 1.414 as the efficiency index. By taking into account the optimality concept many authors have tried to build iterative methods of optimal higher order of convergence. This paper is organized as follows: in section 2, we describe the new third-order iterative method by using the concept of inverse function theorem. In the last section we give some physical example and our new methods are compared in the performance with some well known methods
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