A Family of Multiple-Root Finding Iterative Methods Based on Weight Functions

Francisco I Chicharro,Rafael A Contreras,Neus Garrido

doi:10.3390/math8122194

Francisco I Chicharro, Rafael A Contreras + Show 1 more

Open Access

https://doi.org/10.3390/math8122194

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Journal: Mathematics	Publication Date: Dec 9, 2020
Citations: 7	License type: CC BY 4.0

Affiliation: Universidad Internacional De La Rioja

Abstract

A straightforward family of one-point multiple-root iterative methods is introduced. The family is generated using the technique of weight functions. The order of convergence of the family is determined in its convergence analysis, which shows the constraints that the weight function must satisfy to achieve order three. In this sense, a family of iterative methods can be obtained with a suitable design of the weight function. That is, an iterative algorithm that depends on one or more parameters is designed. This family of iterative methods, starting with proper initial estimations, generates a sequence of approximations to the solution of a problem. A dynamical analysis is also included in the manuscript to study the long-term behavior of the family depending on the parameter value and the initial guess considered. This analysis reveals the good properties of the family for a wide range of values of the parameter. In addition, a numerical test on academic and engineering multiple-root functions is performed.

Highlights

Solving nonlinear equation f ( x ) = 0 has been a difficult problem to handle for a long time in several branches of Science and Engineering
In this paper we focus on a simple idea for generating a class of one-step iterative methods for finding multiple roots of nonlinear equations
This analysis is more common for single root methods; there are papers that include it for multiple root methods [17,18,19,20,21]

Summary

Introduction

Newton’s scheme (or Schröder’s method [15]), based on applying Newton’s iterative scheme to the f (x) functions u1 ( x ) = f ( x )1/m and u2 ( x ) = f 0 ( x) , respectively In both cases, the problem becomes finding a simple root of a nonlinear equation, so the order of convergence is again quadratic. We will be able to select the members of the family whose basins of attraction are wider, ensuring the convergence of the method for a wide set of initial estimates This analysis is more common for single root methods; there are papers that include it for multiple root methods [17,18,19,20,21].

Convergence Analysis of the Parametric Family

Dynamical Analysis of the Family

Preliminaries on Complex Dynamics

The Rational Function

Asymptotic Behavior of Fixed and Critical Points

Numerical Test

Conclusions