Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2014, Special Issue (2014), Article ID 591638, 9 pages.

Optimal High-Order Methods for Solving Nonlinear Equations

S. Artidiello, A. Cordero, Juan R. Torregrosa, and M. P. Vassileva

Full-text: Open access

Abstract

A class of optimal iterative methods for solving nonlinear equations is extended up to sixteenth-order of convergence. We design them by using the weight function technique, with functions of three variables. Some numerical tests are made in order to confirm the theoretical results and to compare the new methods with other known ones.

Article information

Source
J. Appl. Math., Volume 2014, Special Issue (2014), Article ID 591638, 9 pages.

Dates
First available in Project Euclid: 1 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1412178029

Digital Object Identifier
doi:10.1155/2014/591638

Mathematical Reviews number (MathSciNet)
MR3208631

Citation

Artidiello, S.; Cordero, A.; Torregrosa, Juan R.; Vassileva, M. P. Optimal High-Order Methods for Solving Nonlinear Equations. J. Appl. Math. 2014, Special Issue (2014), Article ID 591638, 9 pages. doi:10.1155/2014/591638. https://projecteuclid.org/euclid.jam/1412178029


Export citation

References

  • A. M. Ostrowski, Solution of Equations and System of Equations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1964.
  • 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.
  • M. S. Petković, B. Neta, L. D. Vladimir Petković, and J. Džunić, Multipoint Methods for Solving Nonlinear Equations, Elsevier, New York, NY, USA, 2013.
  • S. Artidiello, F. Chicharro, A. Cordero, and J. R. Torregrosa, “Local convergence and dynamical analysis of a new family of optimal fourth-order iterative methods,” International Journal of Computer Mathematics, vol. 90, no. 10, pp. 2049–2060, 2013.
  • C. Chun, M. Y. Lee, B. Neta, and J. Džunić, “On optimal fourth-order iterative methods free from second derivative and their dynamics,” Applied Mathematics and Computation, vol. 218, no. 11, pp. 6427–6438, 2012.
  • Y. I. Kim, “A triparametric family of three-step optimal eighth-order methods for solving nonlinear equations,” International Journal of Computer Mathematics, vol. 89, no. 8, pp. 1051–1059, 2012.
  • Y. Khan, M. Fardi, and K. Sayevand, “A new general eighth-order family of iterative methods for solving nonlinear equations,” Applied Mathematics Letters, vol. 25, no. 12, pp. 2262–2266, 2012.
  • J. Džunić and M. S. Petković, “A family of three-point methods of Ostrowski's type for solving nonlinear equations,” Journal of Applied Mathematics, vol. 2012, Article ID 425867, 9 pages, 2012.
  • F. Soleymani, M. Sharifi, and B. S. Mousavi, “An improvement of Ostrowski's and King's techniques with optimal convergence order eight,” Journal of Optimization Theory and Applications, vol. 153, no. 1, pp. 225–236, 2012.
  • R. Thukral, “New sixteenth-order derivative-free methods for solving nonlinear equations,” The American Journal of Computational and Applied Mathematics, vol. 2, no. 3, pp. 112–118, 2012.
  • J. R. Sharma, R. K. Guha, and P. Gupta, “Improved King's methods with optimal order of convergence based on rational approximations,” Applied Mathematics Letters, vol. 26, no. 4, pp. 473–480, 2013.
  • J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, New York, NY, USA, 1964.
  • C. Chun, “Some fourth-order iterative methods for solving nonlinear equations,” Applied Mathematics and Computation, vol. 195, no. 2, pp. 454–459, 2008.
  • R. F. King, “A family of fourth order methods for nonlinear equations,” SIAM Journal on Numerical Analysis, vol. 10, pp. 876–879, 1973.
  • L. Zhao, X. Wang, and W. Guo, “New families of eighth-order methods with high efficiency index for solving nonlinear equations,” WSEAS Transactions on Mathematics, vol. 11, pp. 283–293, 2012.
  • J. Džunić, M. S. Petković, and L. D. Petković, “A family of optimal three-point methods for solving nonlinear equations using two parametric functions,” Applied Mathematics and Computation, vol. 217, no. 19, pp. 7612–7619, 2011.
  • 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.
  • P. R. Escobal, Methods of Orbit Determination, Robert E. Krieger Publishing Company, 1965.
  • 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. \endinput