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Submitted
September 18, 2023
Accepted
September 11, 2024
Published
September 23, 2024
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Abstract




A new free derivative iterative method is presented in this article. The method is developed by combining Newton’s method and Euler’s method. Deriva- tives in this method are approximated by forward difference, hyperbola and divided difference. The order of convergence is proven analytically to be of sixth order. Numerical results exhibit that the new method is comparable to other methods. Basins of attraction are also provided to support the proposed method.




Keywords

forward difference divided difference efficiency index derivative-free method order of convergence basins of attraction

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This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

How to Cite
Syamsudhuha, S., Imran, M., Putri, A., Deswita, L., & Amelia, R. (2024). A NEW THREE- STEP DERIVATIVE FREE ITERATIVE METHOD AND ITS DYNAMICS. Journal of the Indonesian Mathematical Society, 30(3), 361–373. https://doi.org/10.22342/jims.30.3.1533.361-373
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References

  1. R. G. Bartle and R. D. Sherbert, Introduction to Real Analysis, 4th ed., John Wiley & Sons, Inc., New York, 2011.
  2. K. E. Atkinson, Elementary Numerical Analysis, 3rd ed., John Wiley & Sons, Inc., New York, 2004.
  3. W. Gautschi, Numerical Analysis, 2nd ed., Birkhauser, New York, 2012.
  4. R. L. Burden and J. D. Faires, Numerical Analysis, 9th ed., Brooks/Cole, Boston, 2001.
  5. J. M. Gutierrez and M. A.Hernandez, ”A Family of Chebyshev-Halley Type Methods in Banach Spaces”. Bulletin Aust. Math. Soc. 55 (1997), 113–130.
  6. Z. Xiaojian, ”Modified Chebyshev-Halley Methods Free from Second Derivative”, Applied Mathematics and Computation 203(2008), 824–827
  7. A. Melman, ”Geometry and Convergence of Euler’s and Halley’s methods”, SIAM Rev. 39(4)(1997), 728-735.
  8. J. R. Sharma, ”A Family of Third-Order Methods to Solve Nonlinear Equations by Quadratic Curves Approximation”, Applied Mathematics and Computation 184(2007), 210-215.
  9. T. R. Scavo and J. B. Thoo, ”On the Geometry of Halley’s method”, Amer. Math. Monthly 102 (1995), 417–426.
  10. A. Syakir, M. Imran, and M. D. H. Gamal, ”Combination of Newton-Halley-Chebyshev Iterative Methods without Second Derivatives”, International Journal of Theoretical and Applied Mathematics 3(3)(2017), 106–109.
  11. R. Amelia, M. Imran, and S. Syamsudhuha, ”Derivative-Free Two Step Iterative Method using Central Difference”, Bulletin of Mathematics 8(01)(2016), 9–17.
  12. J. H. Mathews, Numerical Methods for Mathematics Science and Engineering, 2nd ed., Prentice-Hall International Inc., New Jersey, 1987.
  13. J. F. Traub, Iterative Methods for the Solution of Equations, Prentice Hall, Englewood Cliffs, 1964
  14. Z. Liu and Q. Zheng, ”A Variant of Steffensen’s Method Fourth-Order Convergence and Its Applications”, Applied Mathematics and Computation, 216(2010), 1978–1983.
  15. H. Ren, Q. Wu, and W. Bi, ”A class of Two-Step Steffensen Type Methods with Fourth-Order Convergence”, Applied Mathematics and Computation 209(2009), 206–210.
  16. S. Putra, M. Imran, A. Putri, and R. Marjulisa, ”Dynamic Comparison of Variations of Newton’s Methods with Differerent Types of Mean for Solving Nonlinear Equations”, Int. Journal of Mathematics and Computer Research 10(11) (2022), 2969–2974.
  17. O. S. Solaiman and I. Hashim, ”Efficacy of Optimal Methods for Nonlinear Equations with Chemical Engineering Applications”, Mathematical Problems in Engineering (2019).
  18. C. Chun and B. Neta, ”Comparative Study of Eighth-Order Methods for Finding Simple Roots of Nonlinear Equations”, Numerical Algorithms 74(2017), 1169–1201.
  19. B. Neta and C. Chun, ”Basins of Attraction for Several Optimal Fourth Order Methods for Multiple Roots”, Mathematics and Computers in Simulation 103 (2014),39–59.
  20. B. Neta, C. Chun, and M. Scott, ”Basins of Attraction for Optimal Eighth Order Methods to Find Simple Roots of Nonlinear Equations”, Applied Mathematics and Computation, 227 (2014), 567–592.
  21. B. Neta, M. Scott, and C. Chun, ”Basins of Attraction for Several Methods to Find Simple Roots of Nonlinear Equations”, Applied Mathematics and Computation 218(21)(2012), 10548–10556.
  22. M. Scott, B. Neta ,and C. Chun, ”Basin Attractors for Various Methods”, Applied Mathematics and Computation 218(6)(2011), 2584–2599.