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Complexity
Volume 2017, Article ID 2713145, 15 pages
https://doi.org/10.1155/2017/2713145
Research Article

King-Type Derivative-Free Iterative Families: Real and Memory Dynamics

1Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, València, Spain
2Instituto Tecnológico de Santo Domingo (INTEC), Santo Domingo, Dominican Republic

Correspondence should be addressed to M. P. Vassileva; od.ude.cetni@avoknep.airam

Received 29 June 2017; Accepted 4 October 2017; Published 31 October 2017

Academic Editor: Guido Caldarelli

Copyright © 2017 F. I. Chicharro 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. F. Chicharro, A. Cordero, J. . Gutiérrez, and J. R. Torregrosa, “Complex dynamics of derivative-free methods for nonlinear equations,” Applied Mathematics and Computation, vol. 219, no. 12, pp. 7023–7035, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  2. F. I. Chicharro, A. Cordero, and J. R. Torregrosa, “Dynamics and fractal dimension of Steffensen-type methods,” Algorithms, vol. 8, no. 2, pp. 271–279, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. J. L. Hueso, E. Martínez, and C. Teruel, “Derivative free iterative methods for nonlinear systems,” Applied Mathematics and Computation, vol. 259, pp. 955–966, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  4. Á. A. Magreñán, “A new tool to study real dynamics: the convergence plane,” Applied Mathematics and Computation, vol. 248, pp. 215–224, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  5. Á. A. Magreñán, A. Cordero, J. M. Gutiérrez, and J. R. Torregrosa, “Real qualitative behavior of a fourth-order family of iterative methods by using the convergence plane,” Mathematics and Computers in Simulation, vol. 105, pp. 49–61, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  6. H. T. Kung and J. F. Traub, “Optimal order of one-point and multipoint iteration,” Journal of the ACM, vol. 21, pp. 643–651, 1974. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. A. M. Ostrowski, Solution of Equations and Systems of Equations, Pure and Applied Mathematics, Vol. IX. Academic Press, New York, NY, USA, 1960. View at MathSciNet
  8. J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, New York, NY, USA, 1964. View at MathSciNet
  9. M. S. Petković, B. Neta, L. D. Petković, and J. Džunić, Multipoint Methods for Solving Nonlinear Equations, Academic Press, Amsterdam, The Netherlands, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  10. M. S. Petkovic', B. Neta, L. D. Petkovic', and J. Džunic', “Multipoint methods for solving nonlinear equations: a survey,” Applied Mathematics and Computation, vol. 226, pp. 635–660, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  11. B. Campos, A. Cordero, J. R. Torregrosa, and P. Vindel, “A multidimensional dynamical approach to iterative methods with memory,” Applied Mathematics and Computation, vol. 271, pp. 701–715, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. J. M. Ortega and W. C. Rheinboldt, Solutions of equations and systems of equations, Academic Press, New York, NY, USA, 1960.
  13. P. Blanchard, “Complex analytic dynamics on the Riemann sphere,” Bulletin of the American Mathematical Society, vol. 11, no. 1, pp. 85–141, 1984. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, Redwood, Calif, USA, 1st edition, 1989. View at MathSciNet
  15. J. M. Gutiérrez, S. Plaza, and N. Romero, “Dynamics of a fifth-order iterative method,” International Journal of Computer Mathematics, vol. 89, no. 6, pp. 822–835, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  16. F. Chicharro, A. Cordero, and J. R. Torregrosa, “Drawing dynamical and parameter planes of iterative families and methods,” The Scientific World Journal, vol. 2013, Article ID 780153, 11 pages, 2013. View at Publisher · View at Google Scholar
  17. B. Campos, A. Cordero, J. R. Torregrosa, and P. Vindel, “Stability of king's family of iterative methods with memory,” Journal of Computational and Applied Mathematics, vol. 318, pp. 504–514, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  18. R. C. Robinson, An Introduction to Dynamical Systems—Continuous and Discrete, vol. 19 of Pure and Applied Undergraduate Texts, American Mathematical Society, Providence, Second edition, 2012. View at MathSciNet
  19. J. L. Varona, “Graphic and numerical comparison between iterative methods,” The Mathematical Intelligencer, vol. 24, no. 1, pp. 37–46, 2002. View at Publisher · View at Google Scholar · View at MathSciNet