Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2014, Article ID 737305, 7 pages
http://dx.doi.org/10.1155/2014/737305
Research Article

A New Biparametric Family of Two-Point Optimal Fourth-Order Multiple-Root Finders

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

Received 21 February 2014; Revised 17 June 2014; Accepted 20 June 2014; Published 14 September 2014

Academic Editor: Juan R. Torregrosa

Copyright © 2014 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. L. B. Rall, “Convergence of the Newton process to multiple solutions,” Numerische Mathematik, vol. 9, pp. 23–37, 1966. View at Publisher · View at Google Scholar · View at MathSciNet
  2. H. T. Kung and J. F. Traub, “Optimal order of one-point and multipoint iteration,” Journal of the Association for Computing Machinery, vol. 21, pp. 643–651, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  3. P. Jarratt, “Some efficient fourth order multipoint methods for solving equations,” BIT, vol. 9, pp. 119–124, 1969. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. 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 Google Scholar
  5. 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 MathSciNet · View at Scopus
  6. Y. I. Kim and Y. H. Geum, “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
  7. A. N. Johnson and B. Neta, “High-order nonlinear solver for multiple roots,” Computers & Mathematics with Applications, vol. 55, no. 9, pp. 2012–2017, 2008. View at Publisher · View at Google Scholar
  8. V. Kanwar, S. Bhatia, and M. Kansal, “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 · View at MathSciNet · View at Scopus
  9. X. Li, C. Mu, J. Ma, and L. Hou, “Fifth-order iterative method for finding multiple roots of nonlinear equations,” Numerical Algorithms, vol. 57, no. 3, pp. 389–398, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. 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 MathSciNet · View at Scopus
  11. M. Petkovic, L. Petkovic, and J. Dzunic, “Accelerating generators of iterative methods for finding multiple roots of nonlinear equations,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2784–2793, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  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. J. R. Sharma and R. Sharma, “Modified Chebyshev-Halley type method and its variants for computing multiple roots,” Numerical Algorithms, vol. 61, no. 4, pp. 567–578, 2012. View at Publisher · View at Google Scholar · 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 MathSciNet · View at Scopus
  15. 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 MathSciNet · View at Scopus
  16. 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 MathSciNet · View at Scopus
  17. J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1964. View at MathSciNet
  18. B. Neta, Numerical Methods for the Solution of Equations, Net-A-Sof, Monterey, Calif, USA, 1983.
  19. S. Wolfram, The Mathematica Book, Cambridge University Press, Cambridge, UK, 4th edition, 1999.