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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.

Linked References

  1. S. Amat, S. Busquier, Á. Grau, and M. Grau-Sánchez, “Maximum efficiency for a family of Newton-like methods with frozen derivatives and some applications,” Applied Mathematics and Computation, vol. 219, no. 15, pp. 7954–7963, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  2. M. Hurley and C. Martin, “Newton's algorithm and chaotic dynamical systems,” SIAM Journal on Mathematical Analysis, vol. 15, no. 2, pp. 238–252, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. Hurley, “Attracting orbits in Newton's method,” Transactions of the American Mathematical Society, vol. 297, no. 1, pp. 143–158, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. Cordero, J. R. Torregrosa, and P. Vindel, “Dynamics of a family of Chebyshev-Halleytype methods,” Applied Mathematics and Computation, vol. 219, no. 16, pp. 8568–8583, 2013. View at Publisher · View at Google Scholar
  5. J. M. Gutiérrez, M. A. Hernández, and N. Romero, “Dynamics of a new family of iterative processes for quadratic polynomials,” Journal of Computational and Applied Mathematics, vol. 233, no. 10, pp. 2688–2695, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. G. Honorato, S. Plaza, and N. Romero, “Dynamics of a higher-order family of iterative methods,” Journal of Complexity, vol. 27, no. 2, pp. 221–229, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. S. Plaza and N. Romero, “Attracting cycles for the relaxed Newton's method,” Journal of Computational and Applied Mathematics, vol. 235, no. 10, pp. 3238–3244, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. W. Yang, “Symmetries of the Julia sets of Newton's method for multiple root,” Applied Mathematics and Computation, vol. 217, no. 6, pp. 2490–2494, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. F. A. Potra and V. Pták, Nondiscrete Induction and Iterative Processes, vol. 103 of Research Notes in Mathematics, Pitman, Boston, Mass, USA, 1984. View at MathSciNet
  10. S. Amat, S. Busquier, and S. Plaza, “Chaotic dynamics of a third-order Newton-type method,” Journal of Mathematical Analysis and Applications, vol. 366, no. 1, pp. 24–32, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet