Development of a Family of Jarratt-Like Sixth-Order Iterative Methods for Solving Nonlinear Systems with Their Basins of Attraction

Min-Young Lee,Young Ik Kim

doi:10.3390/a13110303

Abstract

We develop a family of three-step sixth order methods with generic weight functions employed in the second and third sub-steps for solving nonlinear systems. Theoretical and computational studies are of major concern for the convergence behavior with applications to special cases of rational weight functions. A number of numerical examples are illustrated to confirm the convergence behavior of local as well as global character of the proposed and existing methods viewed through the basins of attraction.

Highlights

Since exact solutions for nonlinear equations are rarely available, we usually resort to their numerical solutions
For unified analysis to be performed in both scalar and vector functions, we aim to develop a family of Jarratt-like sixth-order iterative methods by maintaining the same form of weight functions with two derivatives as well as two functional values
A family of Jarratt-like iterative methods for scalar and vector equations is developed and its convergence properties are theoretically established through Theorems 1 and 2

Summary

Introduction

Since exact solutions for nonlinear equations are rarely available, we usually resort to their numerical solutions. −1 where s = f 0 ( xn ) f 0 (yn ), γ is a parameter to be determined later and T f , L f : C → C are weight functions being analytic [16,17,18] in a neighborhood of 1. We are certainly able to introduce generic weight functions using one derivative and three functional values to develop general optimal eighth-order methods that covers the existing ones for the zero of a given scalar function. Expanding such approach to a nonlinear system requires different weight functions. For unified analysis to be performed in both scalar and vector functions, we aim to develop a family of Jarratt-like sixth-order iterative methods by maintaining the same form of weight functions with two derivatives as well as two functional values

Objectives

Methods

Conclusion