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Discrete Dynamics in Nature and Society
Volume 2015, Article ID 378517, 12 pages
http://dx.doi.org/10.1155/2015/378517
Research Article

On Constructing Two-Point Optimal Fourth-Order Multiple-Root Finders with a Generic Error Corrector and Illustrating Their Dynamics

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

Received 2 September 2015; Accepted 15 October 2015

Academic Editor: Alicia Cordero

Copyright © 2015 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.

Linked References

  1. C. Chun and Y. Ham, “Some sixth-order variants of Ostrowski root-finding methods,” Applied Mathematics and Computation, vol. 193, no. 2, pp. 389–394, 2007. View at Publisher · View at Google Scholar · View at Scopus
  2. M. Dehghan and M. Hajarian, “Some derivative free quadratic and cubic convergence iterative formulas for solving nonlinear equations,” Computational and Applied Mathematics, vol. 29, no. 1, pp. 19–30, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  3. H. T. Kung and J. F. Traub, “Optimal order of one-point and multipoint iteration,” Journal of the ACM, vol. 21, no. 4, pp. 643–651, 1974. View at Publisher · View at Google Scholar · View at Scopus
  4. L. B. Rall, “Convergence of the newton process to multiple solutions,” Numerische Mathematik, vol. 9, no. 1, pp. 23–37, 1966. View at Publisher · View at Google Scholar · View at Scopus
  5. J. Traub, Iterative Methods for the Solution of Equations, Chelsea Publishing Company, 1997.
  6. Y. H. Geum and Y. I. Kim, “Cubic convergence of parameter-controlled Newton-secant method for multiple zeros,” Journal of Computational and Applied Mathematics, vol. 233, no. 4, pp. 931–937, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  7. Y. H. Geum and Y. I. Kim, “A two-parameter family of fourth-order iterative methods with optimal convergence for multiple zeros,” Journal of Applied Mathematics, vol. 2013, Article ID 369067, 7 pages, 2013. View at Publisher · View at Google Scholar · View at Scopus
  8. Y. I. Kim and Y. H. Geum, “A new biparametric family of two-point optimal fourth-order multiple-root finders,” Journal of Applied Mathematics, vol. 2014, Article ID 737305, 7 pages, 2014. View at Publisher · View at Google Scholar
  9. C. Dong, “A family of multipoint iterative functions for finding multiple roots of equations,” International Journal of Computer Mathematics, vol. 21, no. 3-4, pp. 363–367, 1987. View at Publisher · View at Google Scholar
  10. M. S. Petković, L. D. Petković, and J. Džunić, “Accelerating generators of iterative methods for finding multiple roots of nonlinear equations,” Computers and Mathematics with Applications, vol. 59, no. 8, pp. 2784–2793, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  11. D. Sbibih, A. Serghini, A. Tijini, and A. Zidna, “A general family of third order method for finding multiple roots,” Applied Mathematics and Computation, vol. 233, pp. 338–350, 2014. View at Publisher · View at Google Scholar · View at Scopus
  12. F. Soleymani and D. K. R. Babajee, “Computing multiple zeros using a class of quartically convergent methods,” Alexandria Engineering Journal, vol. 52, no. 3, pp. 531–541, 2013. View at Publisher · View at Google Scholar · View at Scopus
  13. V. Kanwar, S. Bhatia, and V. Kanwar, “New optimal class of higher-order methods for multiple roots, permitting f´xn=0,” Applied Mathematics and Computation, vol. 222, pp. 564–574, 2013. View at Publisher · View at Google Scholar
  14. X. Zhou, X. Chen, and Y. Song, “Families of third and fourth order methods for multiple roots of nonlinear equations,” Applied Mathematics and Computation, vol. 219, no. 11, pp. 6030–6038, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  15. P. Jarratt, “Some efficient fourth order multipoint methods for solving equations,” BIT Numerical Mathematics, vol. 9, no. 2, pp. 119–124, 1969. View at Publisher · View at Google Scholar · View at Scopus
  16. L. V. Ahlfors, Complex Analysis, McGraw-Hill Book, 1979.
  17. B. V. Shabat, Introduction to Complex Analysis PART II, Functions of Several Variables, American Mathematical Society, 1992.
  18. S. Wolfram, The Mathematica Book, Wolfram Media, 5th edition, 2003.
  19. A. Cordero, J. García-Maimó, J. R. Torregrosa, M. P. Vassileva, and P. Vindel, “Chaos in King's iterative family,” Applied Mathematics Letters, vol. 26, no. 8, pp. 842–848, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  20. R. Behl, A. Cordero, S. S. Motsa, and J. R. Torregrosa, “On developing fourth-order optimal families of methods for multiple roots and their dynamics,” Applied Mathematics and Computation, vol. 265, pp. 520–532, 2015. View at Publisher · View at Google Scholar
  21. S. Amat, S. Busquier, and S. Plaza, “Review of some iterative root-finding methods from a dynamical point of view,” Sientia, vol. 10, pp. 3–35, 2004. View at Google Scholar
  22. T. De Carvalho and M. A. Teixeira, “Basin of attraction of a cusp-fold singularity in 3D piecewise smooth vector fields,” Journal of Mathematical Analysis and Applications, vol. 418, no. 1, pp. 11–30, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  23. S. Amat, S. Busquier, and S. Plaza, “Dynamics of the King and Jarratt iterations,” Aequationes Mathematicae, vol. 69, no. 3, pp. 212–223, 2005. View at Publisher · View at Google Scholar · View at Scopus
  24. I. K. Argyros and Á. A. Magreñán, “On the convergence of an optimal fourth-order family of methods and its dynamics,” Applied Mathematics and Computation, vol. 252, pp. 336–346, 2015. View at Publisher · View at Google Scholar · View at Scopus
  25. F. Chicharro, A. Cordero, J. M. 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 Scopus
  26. C. Chun, M. Y. Lee, B. Neta, and J. Džunić, “On optimal fourth-order iterative methods free from second derivative and their dynamics,” Applied Mathematics and Computation, vol. 218, no. 11, pp. 6427–6438, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  27. C. Chun and B. Neta, “Basins of attraction for several optimal fourth order methods for multiple roots,” Mathematics and Computers in Simulation, vol. 103, pp. 39–59, 2014. View at Publisher · View at Google Scholar · View at Scopus
  28. C. Chun and B. Neta, “Basins of attraction for Zhou-Chen-Song fourth order family of methods for multiple roots,” Mathematics and Computers in Simulation, vol. 109, pp. 74–91, 2015. View at Publisher · View at Google Scholar · View at Scopus
  29. J. A. Wright, J. Deane, M. Bartuccelli, and G. Gentile, “Basins of attraction in forced systems with time-varying dissipation,” Communications in Nonlinear Science and Numerical Simulation, vol. 29, no. 1–3, pp. 72–87, 2015. View at Publisher · View at Google Scholar