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Basins of Attraction for Various Steffensen‐Type Methods

Author

Listed:
  • Alicia Cordero
  • Fazlollah Soleymani
  • Juan R. Torregrosa
  • Stanford Shateyi

Abstract

The dynamical behavior of different Steffensen‐type methods is analyzed. We check the chaotic behaviors alongside the convergence radii (understood as the wideness of the basin of attraction) needed by Steffensen‐type methods, that is, derivative‐free iteration functions, to converge to a root and compare the results using different numerical tests. We will conclude that the convergence radii (and the stability) of Steffensen‐type methods are improved by increasing the convergence order. The computer programming package MATHEMATICA provides a powerful but easy environment for all aspects of numerics. This paper puts on show one of the application of this computer algebra system in finding fixed points of iteration functions.

Suggested Citation

  • Alicia Cordero & Fazlollah Soleymani & Juan R. Torregrosa & Stanford Shateyi, 2014. "Basins of Attraction for Various Steffensen‐Type Methods," Journal of Applied Mathematics, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnljam:v:2014:y:2014:i:1:n:539707
    DOI: 10.1155/2014/539707
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    References listed on IDEAS

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    1. Alicia Cordero & José L. Hueso & Eulalia Martínez & Juan R. Torregrosa, 2012. "A Family of Derivative‐Free Methods with High Order of Convergence and Its Application to Nonsmooth Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    2. Alicia Cordero & José L. Hueso & Eulalia Martínez & Juan R. Torregrosa, 2012. "A Family of Derivative-Free Methods with High Order of Convergence and Its Application to Nonsmooth Equations," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-15, May.
    3. F. Soleymani & S. Karimi Vanani & M. Jamali Paghaleh, 2012. "A Class of Three‐Step Derivative‐Free Root Solvers with Optimal Convergence Order," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    4. F. Soleymani & S. Karimi Vanani & M. Jamali Paghaleh, 2012. "A Class of Three-Step Derivative-Free Root Solvers with Optimal Convergence Order," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-15, February.
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    Cited by:

    1. F. Soleymani & M. Sharifi & S. Shateyi & F. Khaksar Haghani, 2014. "A Class of Steffensen‐Type Iterative Methods for Nonlinear Systems," Journal of Applied Mathematics, John Wiley & Sons, vol. 2014(1).

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