Eighth-Order Iterative Methods without Derivatives for Solving Nonlinear Equations

R Thukral

doi:10.5402/2011/693787

Abstract

A new family of eighth-order derivative-free methods for solving nonlinear equations is presented. It is proved that these methods have the convergence order of eight. These new methods are derivative-free and only use four evaluations of the function per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based on evaluations, could achieve optimal convergence order . Thus, we present new derivative-free methods which agree with Kung and Traub conjecture for . Numerical comparisons are made to demonstrate the performance of the methods presented.

Highlights

Consider iterative methods for finding a simple root α of the nonlinear equation f x 0, 1.1 where f : D ⊂ R → R is a scalar function on an open interval D, and it is sufficiently smooth in a neighbourhood of α
We have demonstrated the performance of a new family of eighth-order derivative-free methods
We have examined the effectiveness of the new derivative-free methods by showing the accuracy of the simple root of a nonlinear equation

Summary

Introduction

Consider iterative methods for finding a simple root α of the nonlinear equation f x 0, 1.1 where f : D ⊂ R → R is a scalar function on an open interval D, and it is sufficiently smooth in a neighbourhood of α. It is well known that the techniques to solve nonlinear equations have many applications in science and engineering. A new family of three-point derivative-free methods of the optimal order eight is constructed by combining optimal two-step fourth-order methods and a modified third step. In order to obtain these new derivative-free methods, we replace derivatives with suitable approximations based on divided difference. It is well known that the various methods have been used in order to approximate the derivatives by the Newton interpolation, the Hermite interpolation, the Lagrange interpolation, and ration function 1, 2

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Conclusion