An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior

Tianbao Liu,Xiwen Qin,Qiuyue Li

doi:10.1515/math-2019-0122

Abstract

Abstract In this paper, we derive and analyze a new one-parameter family of modified Cauchy method free from second derivative for obtaining simple roots of nonlinear equations by using Padé approximant. The convergence analysis of the family is also considered, and the methods have convergence order three. Based on the family of third-order method, in order to increase the order of the convergence, a new optimal fourth-order family of modified Cauchy methods is obtained by using weight function. We also perform some numerical tests and the comparison with existing optimal fourth-order methods to show the high computational efficiency of the proposed scheme, which confirm our theoretical results. The basins of attraction of this optimal fourth-order family and existing fourth-order methods are presented and compared to illustrate some elements of the proposed family have equal or better stable behavior in many aspects. Furthermore, from the fractal graphics, with the increase of the value m of the series in iterative methods, the chaotic behaviors of the methods become more and more complex, which also reflected in some existing fourth-order methods.

Highlights

In this paper, we consider iterative methods to nd a simple root α,i.e, f (α) = and f (α) ≠, of a nonlinear equation f (x) =, (1)where f : I ⊂ R → R for an open interval I is a scalar function
In this paper, we derive and analyze a new one-parameter family of modi ed Cauchy method free from second derivative for obtaining simple roots of nonlinear equations by using Padeapproximant
Based on the family of third-order method, in order to increase the order of the convergence, a new optimal fourth-order family of modi ed Cauchy methods is obtained by using weight function

Summary

Introduction

Based on the family of third-order method, in order to increase the order of the convergence, a new optimal fourth-order family of modi ed Cauchy methods is obtained by using weight function. We introduce parametric weight functions and the well-known technique of undetermined coe cients to the family of iterative methods (17) to increase the order of convergence to four.

Results

Conclusion