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Variants of Chebyshev’s Method with Eighth-Order Convergence for Solving Nonlinear Equations

Received: 26 October 2016     Accepted: 7 January 2017     Published: 3 February 2017
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Abstract

This paper develops the variants of Chebyshev’s method by applying Lagrange interpolation and finite difference to eliminate the second derivative appearing in the Chebyshev’s method. The results of this research show that the modified eight-order method has the efficiency index 1.5157. Numerical simulations show that the effectiveness and performance of the new method in solving nonlinear equations are encouraging.

Published in Applied and Computational Mathematics (Volume 5, Issue 6)
DOI 10.11648/j.acm.20160506.13
Page(s) 247-251
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Chebyshev’s Method, Finite Differences, Lagrange Interpolation, Nonlinear Equations, Order of Convergence

References
[1] S. Amat, S. Busquier, J. M. Guiterres and M. A. Hernandes, On the global convergence of Chebyshev's iterative method, Journal of Computational and Applied Mathematics, 220 (2008), 17-21.
[2] R. Behl and V. Kanwar, Variant of Chebyshev's methods with optimal order convergence, Tamsui Oxford Journal of Information and Mathematical Sciences, 29 (2013), 39-53.
[3] V. Candela and A. Marquina, Recurance relations for rational cubic method, Computing, 45 (1990), 355-367.
[4] H. Esmaeili and A. N. Rezaei, A uniparametric family of modifications for Chebyshev's method, Lecturas Matematicas, 33 (2012), 95-106.
[5] J. Jayakumar and P. Jayasilan, Second derivative free modification with parameter for Chebyshev's method, International Journal of Computational Engineering Research, 03 (2013), 38-42.
[6] V. Kanwar, A family of third order multipoint methods for solving nonlinear equations, Applied Mathematics and Computation, 176 (2006), 409-413.
[7] J. Kou, L. Yitian and W. Xiuhua, A uniparametric Chebyshev-type method free from second derivatives, Applied Mathematics and Computation. 179 (2006), 296-300.
[8] J. Kou, L. Yitian and W. Xiuhua, Third-order modification of Newton's method, Journal of Computational and Applied Mathematics, 205 (2007), 1-5.
[9] M. A. Noor, W. A. Khan and A. Husain, A new modifed Halley method without second derivatives for nonlinear equation, Applied Mathematics and Computation, 189 (2007), 1268-1273.
[10] A. Y. Ozban, Some new variants of Newton's method, Applied Mathematics Letters 13 (2004), 87-93.
[11] S. Weerakoon and T. G. I. Fernando, A variant of Newton's methods with accelarated third order convergence, Applied Mathematics Letters, 1 (2000), 87-93.
Cite This Article
  • APA Style

    Muhamad Nizam Muhaijir, M. Imran, Moh Danil Hendry Gamal. (2017). Variants of Chebyshev’s Method with Eighth-Order Convergence for Solving Nonlinear Equations. Applied and Computational Mathematics, 5(6), 247-251. https://doi.org/10.11648/j.acm.20160506.13

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    ACS Style

    Muhamad Nizam Muhaijir; M. Imran; Moh Danil Hendry Gamal. Variants of Chebyshev’s Method with Eighth-Order Convergence for Solving Nonlinear Equations. Appl. Comput. Math. 2017, 5(6), 247-251. doi: 10.11648/j.acm.20160506.13

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    AMA Style

    Muhamad Nizam Muhaijir, M. Imran, Moh Danil Hendry Gamal. Variants of Chebyshev’s Method with Eighth-Order Convergence for Solving Nonlinear Equations. Appl Comput Math. 2017;5(6):247-251. doi: 10.11648/j.acm.20160506.13

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  • @article{10.11648/j.acm.20160506.13,
      author = {Muhamad Nizam Muhaijir and M. Imran and Moh Danil Hendry Gamal},
      title = {Variants of Chebyshev’s Method with Eighth-Order Convergence for Solving Nonlinear Equations},
      journal = {Applied and Computational Mathematics},
      volume = {5},
      number = {6},
      pages = {247-251},
      doi = {10.11648/j.acm.20160506.13},
      url = {https://doi.org/10.11648/j.acm.20160506.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160506.13},
      abstract = {This paper develops the variants of Chebyshev’s method by applying Lagrange interpolation and finite difference to eliminate the second derivative appearing in the Chebyshev’s method. The results of this research show that the modified eight-order method has the efficiency index 1.5157. Numerical simulations show that the effectiveness and performance of the new method in solving nonlinear equations are encouraging.},
     year = {2017}
    }
    

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    AU  - Muhamad Nizam Muhaijir
    AU  - M. Imran
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    AB  - This paper develops the variants of Chebyshev’s method by applying Lagrange interpolation and finite difference to eliminate the second derivative appearing in the Chebyshev’s method. The results of this research show that the modified eight-order method has the efficiency index 1.5157. Numerical simulations show that the effectiveness and performance of the new method in solving nonlinear equations are encouraging.
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Author Information
  • Department of Mathematics, University of Riau, Pekanbaru, Indonesia

  • Department of Mathematics, University of Riau, Pekanbaru, Indonesia

  • Department of Mathematics, University of Riau, Pekanbaru, Indonesia

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