A Family of Optimal Eighth Order Iteration Functions for Multiple Roots and Its Dynamics

Ghulam Mustafa,Moin-ud-Din Junjua,Faiza Akram,Tariq Afzal,Misbah Arshad,Saima Akram

doi:10.1155/2021/5597186

Abstract

In this manuscript, we present a new general family of optimal iterative methods for finding multiple roots of nonlinear equations with known multiplicity using weight functions. An extensive convergence analysis is presented to verify the optimal eighth order convergence of the new family. Some special cases of the family are also presented which require only three functions and one derivative evaluation at each iteration to reach optimal eighth order convergence. A variety of numerical test functions along with some real-world problems such as beam designing model and Van der Waals’ equation of state are presented to ensure that the newly developed family efficiently competes with the other existing methods. The dynamical analysis of the proposed methods is also presented to validate the theoretical results by using graphical tools, termed as the basins of attraction.

Highlights

In case of multiple roots of nonlinear equations, this classical Newton’s method fails to keep the quadratic convergence and drops to linear convergence, when provided a good initial guess x0 near the exact root α
Many researchers have presented optimal fourth order convergent schemes for multiple roots when multiplicity is known in advance [7,8,9,10,11]. ukral [12], Geum et al [13, 14], and Sharma et al [15] presented nonoptimal sixth and seventh order multiple root finding methods. e optimal convergence order is defined by Kung and Traub [16] that a without memory method can accomplish the order of convergence at most 2n− 1 consuming n function or derivative evaluations
Our aim for the presented work is to develop a general family of multiple root finding methods with simple and compact body structures. erefore, with the demand to construct simple and more effective optimal higher order methods for multiple roots, we present a family of optimal eighth order convergent iterative methods. e proposed scheme requires only four-function evaluations per iterative step which satisfies the classical conjecture given by Kung and Traub [16] and falls in the category of optimal methods. e new simple structured scheme is based on univariate and trivariate weight functions in each iterative step

Summary

Introduction

In case of multiple roots of nonlinear equations, this classical Newton’s method fails to keep the quadratic convergence and drops to linear convergence, when provided a good initial guess x0 near the exact root α. Many researchers have presented optimal fourth order convergent schemes for multiple roots when multiplicity is known in advance [7,8,9,10,11]. E optimal convergence order is defined by Kung and Traub [16] that a without memory method can accomplish the order of convergence at most 2n− 1 consuming n function or derivative evaluations. Efficiency index is defined by Ostrowski [17] as, if the order of convergence of an iterative family is r and the total number of function or derivative evaluations per iteration is n, the efficiency index of an iterative scheme is r(1/n). In 2018, Behl et al [18] presented a class of optimal eighth order methods for approximating multiple roots of nonlinear equations with known multiplicity k, given as follows: yζ xζ. Β2sζ ), wζ e univariate weight function G: C ⟶ C and bivariate weight function H: C2 ⟶ C are analytic in a neighborhood of (0)

Objectives

Methods

Conclusion