An Optimal Order Method for Multiple Roots in Case of Unknown Multiplicity

Jai Jaiswal

doi:10.3390/a9010010

Abstract

In the literature, recently, some three-step schemes involving four function evaluations for the solution of multiple roots of nonlinear equations, whose multiplicity is not known in advance, are considered, but they do not agree with Kung–Traub’s conjecture. The present article is devoted to the study of an iterative scheme for approximating multiple roots with a convergence rate of eight, when the multiplicity is hidden, which agrees with Kung–Traub’s conjecture. The theoretical study of the convergence rate is investigated and demonstrated. A few nonlinear problems are presented to justify the theoretical study.

Highlights

We discuss an iterative scheme for calculating the approximate value of a multiple root ξ of a nonlinear equation ψ( x ) = 0, in R
The modified Newton’s scheme [1] is probably the first method for calculating the estimated value of the multiple root when multiplicity m is available in advance, with a convergence rate twice that in some open set around ξ, under suitable regularity assumptions
To get a higher order method, Li et al [23] proposed an iterative scheme for approximating multiple roots of f ( x ) = 0 with fifth-order convergence, by using the transformation f ( x ) =

Summary

Introduction

The problem of finding multiple roots of ψ( x ) = 0 is equivalent to calculating the simple root of f ( x ) = 0 For this substitution, Newton’s method involved the first and second derivatives . To get a higher order method, Li et al [23] proposed an iterative scheme for approximating multiple roots of f ( x ) = 0 with fifth-order convergence, by using the transformation f ( x ) =. The above mentioned fifth- and sixth-order methods involved four function evaluations [ f ( xn ), f ( xn + f ( xn )), f (yn ), f (un )], and by Kung and Traub’s [25] conjecture, the optimal order should be eight Motivated by this theory, we are going to present an optimal eighth-order method for multiple roots in case of unknown multiplicity: to our best knowledge, this is the first method of the optimal eighth order. Its local rate of convergence is proven and supported by numerical testing

Scheme and Analysis of the Local Convergence Rate

Numerical Testing with the Conclusion