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Discrete Dynamics in Nature and Society
Volume 2016, Article ID 8436759, 23 pages
http://dx.doi.org/10.1155/2016/8436759
Research Article

A Triparametric Family of Optimal Fourth-Order Multiple-Root Finders and Their Dynamics

Department of Applied Mathematics, Dankook University, Cheonan 330-714, Republic of Korea

Received 4 September 2015; Revised 26 December 2015; Accepted 4 January 2016

Academic Editor: Juan R. Torregrosa

Copyright © 2016 Young Ik Kim and Young Hee Geum. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the complex dynamics of a triparametric family of optimal fourth-order multiple-root solvers by analyzing their basins of attraction along with extensive study of Möbius conjugacy maps and extraneous fixed points applied to a prototype quadratic polynomial raised to the power of the known integer multiplicity . A uniform grid centered at the origin covering square region is chosen to display the initial points on each basin of attraction according to a coloring scheme based on their orbit behavior. With illustrative basins of attractions applied to various test polynomials and the corresponding statistical data for convergence as well as a number of comparisons made among the listed methods, we confirm our investigation and analysis developed in this paper.