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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 726701, 10 pages
http://dx.doi.org/10.1155/2013/726701
Research Article

Reducing Chaos and Bifurcations in Newton-Type Methods

1Departamento de Matemática Aplicada y Estadística, Universidad de Cartagena, 30202 Cartagena, Spain
2Departamento de Matemáticas y Computación, Universidad de La Rioja, 26004 Logroño, Spain

Received 15 October 2012; Revised 25 February 2013; Accepted 16 April 2013

Academic Editor: Shukai Duan

Copyright © 2013 S. Amat 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

We study the dynamics of some Newton-type iterative methods when they are applied of polynomials degrees two and three. The methods are free of high-order derivatives which are the main limitation of the classical high-order iterative schemes. The iterative schemes consist of several steps of damped Newton's method with the same derivative. We introduce a damping factor in order to reduce the bad zones of convergence. The conclusion is that the damped schemes become real alternative to the classical Newton-type method since both chaos and bifurcations of the original schemes are reduced. Therefore, the new schemes can be utilized to obtain good starting points for the original schemes.