About this Journal Submit a Manuscript Table of Contents
Advances in Numerical Analysis
Volume 2013 (2013), Article ID 252798, 11 pages
http://dx.doi.org/10.1155/2013/252798
Research Article

On Some Efficient Techniques for Solving Systems of Nonlinear Equations

1Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, Punjab 148106, India
2Department of Mathematics, Government Ranbir College, Sangrur, Punjab 148001, India

Received 18 June 2013; Revised 3 September 2013; Accepted 4 September 2013

Academic Editor: Zhangxing Chen

Copyright © 2013 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. A. M. Ostrowski, Solution of Equations and Systems of Equations, Academic Press, New York, NY, USA, 1966.
  2. J. M. Ortega and W. C. Rheinboldt, Iterative Solutions of Nonlinear Equations in Several Variables, Academic Press, New York, NY, USA, 1970.
  3. F. A. Potra and V. Pták, Nondiscrete Induction and Iterarive Processes, Pitman, Boston, Mass, USA, 1984.
  4. C. T. Kelley, Solving Nonlinear Equations with Newton's Method, SIAM, Philadelphia, Pa, USA, 2003.
  5. J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1964.
  6. J. M. Gutiérrez and M. A. Hernández, “A family of chebyshev-halley type methods in banach spaces,” Bulletin of the Australian Mathematical Society, vol. 55, no. 1, pp. 113–130, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  7. M. Palacios, “Kepler equation and accelerated Newton method,” Journal of Computational and Applied Mathematics, vol. 138, no. 2, pp. 335–346, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. S. Amat, S. Busquier, and J. M. Gutiérrez, “Geometric constructions of iterative functions to solve nonlinear equations,” Journal of Computational and Applied Mathematics, vol. 157, no. 1, pp. 197–205, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. M. Frontini and E. Sormani, “Third-order methods from quadrature formulae for solving systems of nonlinear equations,” Applied Mathematics and Computation, vol. 149, no. 3, pp. 771–782, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  10. H. H. H. Homeier, “A modified Newton method with cubic convergence: the multivariate case,” Journal of Computational and Applied Mathematics, vol. 169, no. 1, pp. 161–169, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  11. M. T. Darvishi and A. Barati, “A fourth-order method from quadrature formulae to solve systems of nonlinear equations,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 257–261, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  12. A. Cordero and J. R. Torregrosa, “Variants of Newton's Method using fifth-order quadrature formulas,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 686–698, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  13. M. A. Noor and M. Waseem, “Some iterative methods for solving a system of nonlinear equations,” Computers and Mathematics with Applications, vol. 57, no. 1, pp. 101–106, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. A. Cordero, E. Martínez, and J. R. Torregrosa, “Iterative methods of order four and five for systems of nonlinear equations,” Journal of Computational and Applied Mathematics, vol. 231, no. 2, pp. 541–551, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  15. A. Cordero, J. L. Hueso, E. Martínez, and J. R. Torregrosa, “A modified Newton-Jarratt's composition,” Numerical Algorithms, vol. 55, no. 1, pp. 87–99, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  16. M. Grau-Sánchez, Á. 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 Zentralblatt MATH · View at Scopus
  17. M. Grau-Sánchez, À. Grau, and M. Noguera, “Ostrowski type methods for solving systems of nonlinear equations,” Applied Mathematics and Computation, vol. 218, no. 6, pp. 2377–2385, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  18. J. R. Sharma, R. K. Guha, and R. Sharma, “An efficient fourth order weighted-Newton method for systems of nonlinear equations,” Numerical Algorithms, vol. 62, no. 2, pp. 307–323, 2013. View at Publisher · View at Google Scholar
  19. S. Wolfram, The Mathematica Book, Wolfram Media, Champaign, Ill, USA, 5th edition, 2003.
  20. M. S. Petković, “Remarks on ‘on a general class of multipoint root-finding methods of high computational efficiency’,” SIAM Journal on Numerical Analysis, vol. 49, no. 3, pp. 1317–1319, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  21. L. Fousse, G. Hanrot, V. Lefèvre, P. Pélissier, and P. Zimmermann, “MPFR: a multiple-precision binary floating-point library with correct rounding,” ACM Transactions on Mathematical Software, vol. 33, no. 2, Article ID 1236468, p. 15, 2007. View at Publisher · View at Google Scholar · View at Scopus
  22. http://www.mpfr.org/mpfr-2.1.0/timings.html.