Table of Contents Author Guidelines Submit a Manuscript
Advances in Numerical Analysis
Volume 2014 (2014), Article ID 152187, 11 pages
http://dx.doi.org/10.1155/2014/152187
Research Article

An Efficient Family of Traub-Steffensen-Type Methods for Solving Systems of Nonlinear Equations

1Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, Punjab 148 106, India
2Department of Mathematics, Government Ranbir College, Sangrur, Punjab 148 001, India

Received 6 February 2014; Accepted 11 June 2014; Published 2 July 2014

Academic Editor: Zhangxin Chen

Copyright © 2014 Janak Raj Sharma and Puneet Gupta. 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.
  2. C. T. Kelley, Solving Nonlinear Equations with Newton's Method, SIAM, Philadelphia, Pa, USA, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  3. J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, New Jersey, NJ, USA, 1964.
  4. J. F. Steffensen, “Remarks on iteration,” Skandinavski Aktuarsko Tidskrift, vol. 16, pp. 64–72, 1933. View at Google Scholar
  5. H. Ren, Q. Wu, and W. Bi, “A class of two-step Steffensen type methods with fourth-order convergence,” Applied Mathematics and Computation, vol. 209, no. 2, pp. 206–210, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. Z. Liu, Q. Zheng, and P. Zhao, “A variant of Steffensen's method of fourth-order convergence and its applications,” Applied Mathematics and Computation, vol. 216, no. 7, pp. 1978–1983, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. M. S. Petković, S. Ilić, and J. Džunić, “Derivative free two-point methods with and without memory for solving nonlinear equations,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 1887–1895, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. M. S. Petković and L. D. Petković, “Families of optimal multipoint methods for solving nonlinear equations: a survey,” Applicable Analysis and Discrete Mathematics, vol. 4, no. 1, pp. 1–22, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. M. S. Petković, J. Džunić, and L. D. Petković, “A family of two-point methods with memory for solving nonlinear equations,” Applicable Analysis and Discrete Mathematics, vol. 5, no. 2, pp. 298–317, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  10. Q. Zheng, J. Li, and F. Huang, “An optimal Steffensen-type family for solving nonlinear equations,” Applied Mathematics and Computation, vol. 217, no. 23, pp. 9592–9597, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. J. Dunić and M. S. Petković, “On generalized multipoint root-solvers with memory,” Journal of Computational and Applied Mathematics, vol. 236, no. 11, pp. 2909–2920, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. J. R. Sharma, R. K. Guha, and P. Gupta, “Some efficient derivative free methods with memory for solving nonlinear equations,” Applied Mathematics and Computation, vol. 219, no. 2, pp. 699–707, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. X. Wang and T. Zhang, “A family of Steffensen type methods with seventh-order convergence,” Numerical Algorithms, vol. 62, no. 3, pp. 429–444, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. M. S. Petkovic, B. Neta, L. D. Petkovic, and J. Dzunic, Multipoint Methods for Solving Nonlinear Equations, Elsevier, Boston, Mass, USA, 2013.
  15. M. Grau-Sánchez, À. Grau, and M. Noguera, “Frozen divided difference scheme for solving systems of nonlinear equations,” Journal of Computational and Applied Mathematics, vol. 235, no. 6, pp. 1739–1743, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. M. Grau-Sánchez and M. Noguera, “A technique to choose the most efficient method between secant method and some variants,” Applied Mathematics and Computation, vol. 218, no. 11, pp. 6415–6426, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. M. Grau-Sánchez, À. Grau, and M. Noguera, “On the computational efficiency index and some iterative methods for solving systems of nonlinear equations,” Journal of Computational and Applied Mathematics, vol. 236, no. 6, pp. 1259–1266, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. F. A. Potra and V. Ptak, Nondiscrete Induction and Iterarive Processes, Pitman, Boston, Mass, USA, 1984. View at MathSciNet
  19. S. Wolfram, The Mathematica Book, Wolfram Media, 5th edition, 2003. View at MathSciNet
  20. L. O. Jay, “A note on Q-order of convergence,” BIT Numerical Mathematics, vol. 41, no. 2, pp. 422–429, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. L. Fousse, G. Hanrot, V. Lefevre, P. Pelissier, and P. Zimmermann, “MPFR: a multiple-precision binary oating-point library with correct rounding,” ACM Transactions on Mathematical Software, vol. 33, no. 2, article 13, 15 pages, 2007. View at Publisher · View at Google Scholar
  22. http://www.mpfr.org/mpfr-2.1.0/timings.html.
  23. M. Grau-Sánchez, J. M. Peris, and J. M. Gutiérrez, “Accelerated iterative methods for finding solutions of a system of nonlinear equations,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1815–1823, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. J. A. Ezquerro, M. Grau-Sánchez, and M. Hernández, “Solving non-differentiable equations by a new one-point iterative method with memory,” Journal of Complexity, vol. 28, no. 1, pp. 48–58, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus