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Journal of Applied Mathematics
Volume 2014 (2014), Article ID 539707, 17 pages
http://dx.doi.org/10.1155/2014/539707
Research Article

Basins of Attraction for Various Steffensen-Type Methods

1Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, 46022 Valencia, Spain
2Department of Mathematics, Islamic Azad University, Zahedan Branch, Zahedan, Iran
3Department of Mathematics and Applied Mathematics, School of Mathematical and Natural Sciences, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa

Received 19 December 2013; Revised 31 January 2014; Accepted 31 January 2014; Published 12 March 2014

Academic Editor: Ioannis K. Argyros

Copyright © 2014 Alicia Cordero et al. 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

The dynamical behavior of different Steffensen-type methods is analyzed. We check the chaotic behaviors alongside the convergence radii (understood as the wideness of the basin of attraction) needed by Steffensen-type methods, that is, derivative-free iteration functions, to converge to a root and compare the results using different numerical tests. We will conclude that the convergence radii (and the stability) of Steffensen-type methods are improved by increasing the convergence order. The computer programming package MATHEMATICA provides a powerful but easy environment for all aspects of numerics. This paper puts on show one of the application of this computer algebra system in finding fixed points of iteration functions.