Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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SOME OPTIMAL METHODS WITH EIGHTH-ORDER CONVERGENCE FOR THE SOLUTION OF NONLINEAR EQUATIONS

  • Received : 2016.09.20
  • Accepted : 2016.10.13
  • Published : 2016.11.15
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Abstract

In this paper we propose a new family of eighth order optimal methods for solving nonlinear equations by using weight function methods. The methods of the family require three function and one derivative evaluations per step and has order of convergence eight, and so they are optimal in the sense of Kung-Traub hypothesis. Precise analysis of convergence is given. Some members of the family are compared with several existing methods to show their performance and as a result to confirm that our methods are as competitive as compared to them.

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References

  1. J. F. Traub, Iterative Methods for the Solution of Equations, Chelsea Publishing Company, New York, (1977).
  2. M. S. Petkovic, B. Neta, L. D. Petkovic, and J. Dzunic, Multipoint Methods for Solving Nonlinear Equations, Elsevier, (2012).
  3. H. T. Kung and J. F. Traub, Optimal order of one-point and multipoint iterations, J. Assoc. Comput. Math. 21 (1974), 643-651. https://doi.org/10.1145/321850.321860
  4. A. M. Ostrowski, Solution of Equations and Systems of Equations, Academic Press, New York, London, (1966).
  5. Y. H. Geum and Y. I. Kim, A uniparametric family of three-step eighth-order multipoint iterative methods for simple roots, Appl. Math. Lett. 24 (2011), 929-935. https://doi.org/10.1016/j.aml.2011.01.002
  6. Y. H. Geum and Y. I. Kim, A biparametric family of eighth-order methods with their third-step weighting function decomposed into a one-varaiable linear fraction and a two-variable generic function, Comput. Math. Appl. 61 (2011), 708-714. https://doi.org/10.1016/j.camwa.2010.12.020
  7. A. Cordero, J. R. Torregrosa, and M. P. Vassileva, Three-step iterative methods with optimal-order convergence, J. Comput. Appl. Math. 235 (2011), 3189-3194. https://doi.org/10.1016/j.cam.2011.01.004
  8. A. Cordero, J. R. Torregrosa, and M. P. Vassileva, A family of modfied Ostrowski's methods with optimal eight order of convergence, Appl. Math. Lett. 24 (2011), 2082-2086. https://doi.org/10.1016/j.aml.2011.06.002
  9. J. Dzunic, M. S. Petkovic, and L. D. Petkovic, A family of optimal three-point methods for solving nonlinear equations using two parametric functions, Appl. Math. Comput. 217 (2011), 7612-7619.
  10. J. Dzunic, and M. S. Petkovic, On generalized multipoint root-solvers with memory, J. Comput. Appl. Math. 236 (2012), 2909-2920. https://doi.org/10.1016/j.cam.2012.01.035
  11. C. Chun and B. Neta, An analysis of a King-based family of optimal eighth-order methods, Amer. J. Algorithms and Computing 217 (2015).
  12. C. Chun, Some variants of King's fourth-order family of methods for nonlinear equations, Appl. Math. Comput. 190 (2007), 57-62.
  13. R. Thukral, A new eighth-order iterative method for solving nonlinear equations, Appl. Math. Comput. 217 (2010), 222-229. https://doi.org/10.1016/j.amc.2010.05.048
  14. L. Liu and X.Wang, Eighth-order methods with high efficiency index for solving nonlinear equations, J. Comput. Appl. Math. 215 (2010), 3449-3454. https://doi.org/10.1016/j.amc.2009.10.040
  15. R. Behl, V. Kanwar, and K. K. Sharma, Optimal equi-scaled families of Jarratt's method, Int. J. Comput. Math. 90 (2013), 408-422. https://doi.org/10.1080/00207160.2012.719078
  16. J. R. Sharma, R. K. Guha, and P. Gupta, Improved King's methods with optimal order of convergence based on rational approximations, Appl. Math. Lett. 26 (2013), 473-480. https://doi.org/10.1016/j.aml.2012.11.011
  17. R. Behl, C. Chun, and S. S. Motsa, A general way to construct a new optimal scheme with eighth-order convergence for nonlinear equations, Submitted for publication.
  18. J. Dzunic and M. Petkovic, A family of three-point methods of Ostrowski's type for solving nonlinear equations, J. Appl. Math. 2012 (2012), Article ID 425867, 9 pages, http://dx.doi.org/10.1155/2012/425867.
  19. S. K. K and T. Steihaug, Algorithm for forming derivative-free optimal methods, Numer. Algor. 65 (2014), 809-824. https://doi.org/10.1007/s11075-013-9715-x
  20. F. Soleymani, S. K. Vanani, M. Khan, and M. Sharifi, Some modifications of King's family with optimal eighth-order of convergence, Math. Comput. Model. 55 (2012), 1373-1380. https://doi.org/10.1016/j.mcm.2011.10.016
  21. A. Cordero, J. R. Torregrosa, and M. P. Vassileva, Three-step iterative methods with optimal eighth-order convergence, J. Comput. Appl. Math. 235 (2011), 3189-3194. https://doi.org/10.1016/j.cam.2011.01.004
  22. M. Heydari, S. M. Hosseini, and G. B. Loghmani, On two new families of iterative methods for solving nonlinear equations with optimal order, Appl. Anal. Disc. Math. 5 (2011), 93-109. https://doi.org/10.2298/AADM110228012H