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Journal of Applied Mathematics
Volume 2014, Article ID 812072, 9 pages
http://dx.doi.org/10.1155/2014/812072
Research Article

New Families of Third-Order Iterative Methods for Finding Multiple Roots

1Department of Mathematics, Taizhou University, Linhai, Zhejiang 317000, China
2College of Information and Engineering, Hangzhou Polytechnic, Hangzhou, Zhejiang 311402, China
3Department of Mathematics, Brno University of Technology, Brno, Czech Republic
4Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, China
5Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China

Received 5 February 2014; Revised 19 May 2014; Accepted 20 May 2014; Published 15 June 2014

Academic Editor: Alicia Cordero

Copyright © 2014 R. F. Lin 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. E. Schröder, “Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen,” Mathematische Annalen, vol. 2, no. 2, pp. 317–365, 1870. View at Publisher · View at Google Scholar · View at Scopus
  2. E. Hansen and M. Patrick, “A family of root finding methods,” Numerische Mathematik, vol. 27, no. 3, pp. 257–269, 1976. View at Publisher · View at Google Scholar · View at Scopus
  3. J. F. Traub, Iterative Methods for the Solution of Equations, Chelsea, New York, NY, USA, 1977.
  4. C. Dong, “A basic theorem of constructing an iterative formula of the higher order for computing multiple roots of an equation,” Mathematica Numerica Sinica, vol. 11, pp. 445–450, 1982. View at Google Scholar · View at Zentralblatt MATH
  5. H. D. Victory Jr. and B. Neta, “A higher order method for multiple zeros of nonlinear functions,” International Journal of Computer Mathematics, vol. 12, no. 3-4, pp. 329–335, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  6. C. Dong, “A family of multipoint iterative functions for finding multiple roots of equations,” International Journal of Computer Mathematics, vol. 21, pp. 363–367, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. N. Osada, “An optimal multiple root-finding method of order three,” Journal of Computational and Applied Mathematics, vol. 51, no. 1, pp. 131–133, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. M. Frontini and E. Sormani, “Modified Newton's method with third-order convergence and multiple roots,” Journal of Computational and Applied Mathematics, vol. 156, no. 2, pp. 345–354, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. B. Neta, “New third order nonlinear solvers for multiple roots,” Applied Mathematics and Computation, vol. 202, no. 1, pp. 162–170, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  10. B. Neta and A. N. Johnson, “High-order nonlinear solver for multiple roots,” Computers and Mathematics with Applications, vol. 55, no. 9, pp. 2012–2017, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  11. B. Neta, “Extension of Murakami's high-order non-linear solver to multiple roots,” International Journal of Computer Mathematics, vol. 87, no. 5, pp. 1023–1031, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  12. C. Chun and B. Neta, “A third-order modification of Newton's method for multiple roots,” Applied Mathematics and Computation, vol. 211, no. 2, pp. 474–479, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  13. C. Chun, H. J. Bae, and B. Neta, “New families of nonlinear third-order solvers for finding multiple roots,” Computers and Mathematics with Applications, vol. 57, no. 9, pp. 1574–1582, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. 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 Zentralblatt MATH · View at Scopus
  15. S. Li, H. Li, and L. Cheng, “Some second-derivative-free variants of Halley's method for multiple roots,” Applied Mathematics and Computation, vol. 215, no. 6, pp. 2192–2198, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  16. H. H. H. Homeier, “On Newton-type methods for multiple roots with cubic convergence,” Journal of Computational and Applied Mathematics, vol. 231, no. 1, pp. 249–254, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  17. M. S. Petkovi, J. Duni, and M. Miloevi, “Traub's accelerating generator of iterative root-finding methods,” Applied Mathematics Letters, vol. 24, no. 8, pp. 1443–1448, 2011. View at Publisher · View at Google Scholar · View at Scopus
  18. B. I. Yun, “Iterative methods for solving nonlinear equations with finitely many roots in an interval,” Journal of Computational and Applied Mathematics, vol. 236, no. 13, pp. 3308–3318, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  19. P. Jarratt, “Some fourth order multipoint methods for solving equations,” Mathematics of Computation, vol. 20, pp. 434–437, 1966. View at Google Scholar
  20. 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 Zentralblatt MATH · View at Scopus
  21. 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
  22. B. Neta and C. Chun, “On a family of Laguerre methods to find multiple roots of nonlinear equations,” Applied Mathematics and Computation, vol. 219, no. 23, pp. 10987–11004, 2013. View at Publisher · View at Google Scholar · View at Scopus
  23. D. Sbibiha, 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
  24. B. I. Yun, “Transformation methods for finding multiple roots of nonlinear equations,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 599–606, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  25. R. F. King, “A family of fourth order methods for nonlinear equations,” SIAM Journal on Numerical Analysis, vol. 10, pp. 876–879, 1973. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer, New York, NY, USA, 1980.
  27. S. Weerakoon and T. G. I. Fernando, “A variant of Newton's method with accelerated third-order convergence,” Applied Mathematics Letters, vol. 13, no. 8, pp. 87–93, 2000. View at Google Scholar · View at Zentralblatt MATH · View at Scopus