Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2016, Article ID 8436759, 23 pages
http://dx.doi.org/10.1155/2016/8436759
Research Article

A Triparametric Family of Optimal Fourth-Order Multiple-Root Finders and Their Dynamics

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

Received 4 September 2015; Revised 26 December 2015; Accepted 4 January 2016

Academic Editor: Juan R. Torregrosa

Copyright © 2016 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. J. A. Wright, J. H. 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 · View at MathSciNet
  2. I. K. Argyros and Á. 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 MathSciNet · View at Scopus
  3. 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 MathSciNet · View at Scopus
  4. B. Neta and C. Chun, “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 MathSciNet · View at Scopus
  5. S. Kumar, V. Kanwar, and S. Singh, “On some modified families of multipoint iterative methods for multiple roots of nonlinear equations,” Applied Mathematics and Computation, vol. 218, no. 14, pp. 7382–7394, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. L. Shengguo, L. Xiangke, and C. Lizhi, “A new fourth-order iterative method for finding multiple roots of nonlinear equations,” Applied Mathematics and Computation, vol. 215, no. 3, pp. 1288–1292, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. S. G. Li, L. Z. Cheng, and B. Neta, “Some fourth-order nonlinear solvers with closed formulae for multiple roots,” Computers & Mathematics with Applications, vol. 59, no. 1, pp. 126–135, 2010. View at Publisher · View at Google Scholar · View at Scopus
  8. V. Kanwar, S. Bhatia, and M. Kansal, “New optimal class of higher-order methods for multiple roots, permitting fxn=0,” Applied Mathematics and Computation, vol. 222, no. 1, pp. 564–574, 2013. View at Publisher · View at Google Scholar
  9. 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 MathSciNet · View at Scopus
  10. A. M. Ostrowski, Solutions of Equations and System of Equations, Academic Press, New York, NY, USA, 1960.
  11. M. S. Petković, B. Neta, L. D. Petković, and J. Džunić, Multipoint Methods for Solving Non-Linear Equations, Academic Press, 2012.
  12. J. R. Sharma and R. Sharma, “Modified Jarratt method for computing multiple roots,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 878–881, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. F. Soleymani, D. K. Babajee, and T. Lotfi, “On a numerical technique for finding multiple zeros and its dynamic,” Journal of the Egyptian Mathematical Society, vol. 21, no. 3, pp. 346–353, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  14. X. Zhou, X. Chen, and Y. Song, “Constructing higher-order methods for obtaining the multiple roots of nonlinear equations,” Journal of Computational and Applied Mathematics, vol. 235, no. 14, pp. 4199–4206, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. 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 MathSciNet · View at Scopus
  16. 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 MathSciNet · View at Scopus
  17. 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 Zentralblatt MATH · View at MathSciNet · View at Scopus
  18. L. V. Ahlfors, Complex Analysis, McGraw-Hill Book, 1979. View at MathSciNet
  19. 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 MathSciNet
  20. V. Kanwar, S. Bhatia, and M. Kansal, “New optimal class of higher-order methods for multiple roots, permitting fxn=0,” Applied Mathematics and Computation, vol. 222, pp. 564–574, 2013. View at Google Scholar
  21. 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
  22. 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 MathSciNet
  23. 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 · View at MathSciNet
  24. J. Traub, Iterative Methods for the Solution of Equations, Chelsea, 1997.
  25. S. Wolfram, The Mathematica Book, Wolfram Media, 5th edition, 2003.
  26. A. F. Beardon, Iteration of Rational Functions, Springer, New York, NY, USA, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  27. P. Blanchard, “The dynamics of Newton's method,” Proceedings of Symposia in Applied Mathematics, vol. 49, pp. 139–154, 1994. View at Google Scholar
  28. 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 · View at MathSciNet
  29. A. A. Magrenan, A. Cordero, J. M. Gutierrez, 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 · View at Scopus
  30. A. Cordero, C. Jordán, and J. Torregrosa, “One-point Newton-type iterative methods: a unified point of view,” Journal of Computational and Applied Mathematics, vol. 275, pp. 366–374, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. B. Kalantari and Y. Jin, “On extraneous fixed-points of the basic family of iteration functions,” BIT Numerical Mathematics, vol. 43, no. 2, pp. 453–458, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. E. R. Vrscay and W. J. Gilbert, “Extraneous fixed points, basin boundaries and chaotic dynamics for Schröder and König rational iteration functions,” Numerische Mathematik, vol. 52, no. 1, pp. 1–16, 1988. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. E. Schröder, “Über unendlich viele algorithmen zur auflösung der gleichungen,” Mathematische Annalen, vol. 2, pp. 317–365, 1870. View at Google Scholar
  34. A. Cayley, “Application of the Newton-Fourier method to an imaginary root of an equation,” The Quarterly Journal of Pure and Applied Mathematics, vol. 16, pp. 179–185, 1879. View at Google Scholar
  35. M. Scott, B. Neta, and C. Chun, “Basin attractors for various methods,” Applied Mathematics and Computation, vol. 218, no. 6, pp. 2584–2599, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  36. B. Neta, M. Scott, and C. Chun, “Basins of attraction for several methods to find simple roots of nonlinear equations,” Applied Mathematics and Computation, vol. 218, no. 21, pp. 10548–10556, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus