About this Journal Submit a Manuscript Table of Contents
Advances in Numerical Analysis
Volume 2014 (2014), Article ID 963878, 8 pages
http://dx.doi.org/10.1155/2014/963878
Research Article

General Family of Third Order Methods for Multiple Roots of Nonlinear Equations and Basin Attractors for Various Methods

1Department of Applied Sciences, D.A.V. Institute of Engineering and Technology, Kabir Nagar, Jalandhar 144008, India
2Department of Mathematics, D.A.V. College, Jalandhar 144008, India

Received 29 November 2013; Accepted 8 February 2014; Published 31 March 2014

Academic Editor: Ting-Zhu Huang

Copyright © 2014 Rajni Sharma and Ashu Bahl. 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. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, NY, USA, 1970. View at MathSciNet
  2. S. C. Chapra and R. P. Canale, Numerical Methods for Engineers, McGraw-Hill Book Company, New York, NY, USA, 1988.
  3. 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 MathSciNet
  4. A. M. Ostrowski, Solution of Equations in Euclidean and Banach Spaces, Academic Press, New York, NY, USA, 3rd edition, 1973. View at MathSciNet
  5. E. Hansen and M. Patrick, “A family of root finding methods,” Numerische Mathematik, vol. 27, no. 3, pp. 257–269, 1977. View at Zentralblatt MATH · View at MathSciNet
  6. J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1964. View at MathSciNet
  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 MathSciNet
  8. 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 MathSciNet
  9. 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 MathSciNet
  10. 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 MathSciNet
  11. J. Biazar and B. Ghanbari, “A new third-order family of nonlinear solvers for multiple roots,” Computers and Mathematics with Applications, vol. 59, no. 10, pp. 3315–3319, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. R. Sharma and R. Sharma, “New third and fourth order nonlinear solvers for computing multiple roots,” Applied Mathematics and Computation, vol. 217, no. 23, pp. 9756–9764, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  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 Zentralblatt MATH · View at MathSciNet
  14. 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 MathSciNet
  15. W. Gautschi, Numerical Analysis: An Introduction, Birkhäuser, Boston, Mass, USA, 1997. View at MathSciNet
  16. 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 Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. S. Wolfram, The Mathematica Book, Wolfram Media, Champaign, Ill, USA, 5th edition, 2003. View at Zentralblatt MATH · View at MathSciNet
  18. A. Cayley, “The Newton-Fourier imaginary problem,” The American Journal of Mathematics, vol. 2, no. 1, article 97, 1879. View at Zentralblatt MATH
  19. M. L. Sahari and I. Djellit, “Fractal Newton basins,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 28756, 16 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. 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 Zentralblatt MATH · View at MathSciNet
  21. 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
  22. B. Neta, M. Scott, and C. Chun, “Basin attractors for various methods for multiple roots,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5043–5066, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet