Submit Manuscript

Asian Journal of Scientific Research

Year: 2018 | Volume: 11 | Issue: 3 | Page No.: 409-414
Research Article

A New Variant of Chebyshev-Halley’s Method Without Second Derivative with Convergence of Optimal Order

Wartono
Department of Mathematics, State Islamic University of Sultan Syarif Kasim Riau, 28293 Pekanbaru, Indonesia
M. Soleh
Department of Mathematics, State Islamic University of Sultan Syarif Kasim Riau, 28293 Pekanbaru, Indonesia
I. Suryani
Department of Mathematics, State Islamic University of Sultan Syarif Kasim Riau, 28293 Pekanbaru, Indonesia
Zulakmal
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Andalas, Kampus Unand Limau Manis, 25163 Padang, Indonesia
Muhafzan
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Andalas, Kampus Unand Limau Manis, 25163 Padang, Indonesia
LiveDNA: 62.20729
Background and Objective: The Chebyshev-Halley is an third order iterative method that be used to find the roots of a nonlinear equation. This study is presented a new variant of Chebyshev-Halley’s method without second derivative with two parameters. Methodology: In order to avoid the second derivative, it is approximated by using an equality of two methods, namely, use of a circle of curvature that has the same tangent line and to equate to the Potra-Ptak’s method. Results: The results show that the method requires two evaluation of functions and one its first derivative per iteration with the efficiency index equal to 4⅓ ≈ 1.5874. The convergence analysis shows that the proposed method has the fourth-order convergence for θ = 1 and β = 1 and requires three evaluation of functions per iteration. Conclusion: The final results show that the proposed methods has better performance as compared some other kind of methods. A numerical simulation is presented to show the performance of the proposed method by using several functions.
PDF Fulltext XML References Citation

Leave a Comment

Your email address will not be published. Required fields are marked *

Article Trend

Total views 2405

References

  1. Ostrowski, A.M., 1973. Solution of Equations in Euclidean and Banach Space. Academic Press, New York.
  2. Amat, S., S. Busquier and J.M. Gutierrez, 2003. Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Applied Math., 157: 197-205.
    CrossRefDirect Link
  3. Hernandez, M.A. and M.A. Salanova, 1993. A family of Chebyshev-Halley type methods. Int. J. Comput. Math., 47: 59-63.
    CrossRefDirect Link
  4. Gutierrez, J.M. and M.A. Hernandez, 1997. A family of Chebyshev-Halley type methods in banach spaces. Bull. Aust. Math. Soc., 55: 113-130.
    CrossRefDirect Link
  5. Kanwar, V. and S.K. Tomar, 2007. Modified families of multi-point iterative methods for solving nonlinear equations. Numer. Algorithms, 44: 381-389.
    CrossRefDirect Link
  6. Li, Y., P. Zhang and Y. Li, 2010. Some new variants of Chebyshev-Halley methods free from second derivative. Int. J. Nonlinear Sci., 9: 201-206.
    Direct Link
  7. Nedzhibov, G.H., V.I. Hasanov and M.G. Petkov, 2006. On some families of multi-point iterative methods for solving nonlinear equations. Numer. Algorithms, 42: 127-136.
    CrossRefDirect Link
  8. Wartono, M. Soleh, I. Suryani and Muhafzan, 2016. Chebyshev-Halley's method without second derivative of eight-order convergence. Global J. Pure Applied Math., 12: 2987-2997.
    Direct Link
  9. Chun, C.B., 2007. Certain improvements of Chebyshev-Halley methods with accelerated fourth-order convergence. Applied Math. Comput., 189: 597-601.
    CrossRefDirect Link
  10. Kou, J., Y. Li and X. Wang, 2007. Fourth-order iterative methods free from second derivative. Applied Math. Comput., 184: 880-885.
    CrossRefDirect Link
  11. Rostami, M. and H. Esmaeili, 2014. A modification of Chebyshev-Halley method free from second derivatives for nonlinear equations. Caspian J. Math. Sci., 3: 133-140.
  12. Chun, C., 2007. Some variants of Chebyshev-Halley methods free from second derivative. Applied Math. Comput., 191: 193-198.
    CrossRefDirect Link
  13. Chun, C., 2007. Some second-derivative-free variants of Chebyshev-Halley methods. Applied Math. Comput., 191: 410-414.
    CrossRefDirect Link
  14. Grau-Sanchez, M. and J.M. Gutierrez, 2010. Some variants of the Chebyshev-Halley family of methods with fifth order of convergence. Int. J. Comput. Math., 87: 818-833.
    CrossRefDirect Link
  15. Xiaojian, Z., 2008. Modified Chebyshev-Halley methods free from second derivative. Applied Math. Comput., 203: 824-827.
    CrossRefDirect Link
  16. Chun, C., 2008. A simply constructed third-order modifications of Newton's method. J. Comput. Applied Math., 219: 81-89.
    CrossRefDirect Link
  17. Potra, F.A. and V. Ptak, 1984. Nondiscrete Induction and Iterative Processes. Vol. 103, Pitman Advanced Publisher, Boston, Massachusetts, USA., ISBN:9780273086277, Pages: 207.
  18. Sharma, J.R., 2005. A composite third order Newton-Steffensen method for solving nonlinear equations. Applied Math. Comput., 169: 242-246.
    CrossRefDirect Link
  19. Kung, H.T. and J.F. Traub, 1974. Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Mach., 21: 643-651.
    CrossRefDirect Link
  20. Traub, J.F., 1964. Iterative Methods for the Solution of Equations. Prentince-Hall Inc., New York.

Keywords