Optimal sixteenth order convergent method based on quasi-Hermite interpolation for computing roots.

Fiza Zafar,Nawab Hussain,Zirwah Fatimah,Athar Kharal

doi:10.1155/2014/410410

Abstract

We have given a four-step, multipoint iterative method without memory for solving nonlinear equations. The method is constructed by using quasi-Hermite interpolation and has order of convergence sixteen. As this method requires four function evaluations and one derivative evaluation at each step, it is optimal in the sense of the Kung and Traub conjecture. The comparisons are given with some other newly developed sixteenth-order methods. Interval Newton's method is also used for finding the enough accurate initial approximations. Some figures show the enclosure of finitely many zeroes of nonlinear equations in an interval. Basins of attractions show the effectiveness of the method.

Highlights

Let us consider the problem of approximating the simple root x∗ of the nonlinear equation involving a nonlinear univariate function f: f (x) = 0. (1)Newton’s method and its variants have always remained as widely used one-point without memory and one-step methods for solving (1)
A general four-step four-point iterative method without memory has been given for solving nonlinear equations
This iterative method has been obtained by approximating the first derivative of the function at the fourth step by using quasiHermite interpolation

Summary

Introduction

To overcome the drawbacks of one-point, one-step methods, many multipoint multistep higher order convergent methods have been introduced in the recent past by using inverse, Hermite, and rational interpolation [1, 2]. In developing these methods, so far, the conjecture of Kung and Traub has remained the focus of attention. The conjecture of Kung and Traub has remained the focus of attention An optimal iterative method without memory based on n evaluations would achieve an optimal convergence order of 2n−1, a computational efficiency of 2(n−1)/n

Methods

Results

Conclusion