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Construction of fourth-order optimal families of iterative methods and their dynamics

Author

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  • Behl, Ramandeep
  • Cordero, Alicia
  • Motsa, Sandile S.
  • Torregrosa, Juan R.

Abstract

In this paper, we propose a general class of fourth-order optimal multi-point methods without memory for obtaining simple roots. This class requires only three functional evaluations (viz. two evaluations of function f(xn), f(yn) and one of its first-order derivative f ′(xn)) per iteration. Further, we show that the well-known Ostrowski’s method and King’s family of fourth-order procedures are special cases of our proposed schemes. One of the new particular subclasses is a biparametric family of iterative methods. By using complex dynamics tools, its stability is analyzed, showing stable members of the family. Further on, one of the parameters is fixed and the stability of the resulting class is studied. On the other hand, the accuracy and validity of new schemes is tested by a number of numerical examples by comparing them with recent and classical optimal fourth-order methods available in the literature. It is found that they are very useful in high precision computations.

Suggested Citation

  • Behl, Ramandeep & Cordero, Alicia & Motsa, Sandile S. & Torregrosa, Juan R., 2015. "Construction of fourth-order optimal families of iterative methods and their dynamics," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 89-101.
    • Handle: RePEc:eee:apmaco:v:271:y:2015:i:c:p:89-101
    DOI: 10.1016/j.amc.2015.08.113
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    References listed on IDEAS

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    1. Argyros, Ioannis K. & Magreñán, Á. Alberto, 2015. "On the convergence of an optimal fourth-order family of methods and its dynamics," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 336-346.
    2. Chun, Changbum & Neta, Beny, 2015. "Comparing the basins of attraction for Kanwar–Bhatia–Kansal family to the best fourth order method," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 277-292.
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    Cited by:

    1. Zhanlav, T. & Chuluunbaatar, O. & Ulziibayar, V., 2017. "Generating function method for constructing new iterations," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 414-423.
    2. Petković, I. & Herceg, Ð., 2017. "Symbolic computation and computer graphics as tools for developing and studying new root-finding methods," Applied Mathematics and Computation, Elsevier, vol. 295(C), pages 95-113.
    3. Fiza Zafar & Alicia Cordero & Juan R. Torregrosa, 2018. "An Efficient Family of Optimal Eighth-Order Multiple Root Finders," Mathematics, MDPI, vol. 6(12), pages 1-16, December.
    4. van Lith, Bart S. & ten Thije Boonkkamp, Jan H.M. & IJzerman, Wilbert L., 2018. "Full linear multistep methods as root-finders," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 190-201.
    5. Campos, B. & Vindel, P., 2021. "Dynamics of subfamilies of Ostrowski–Chun methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 181(C), pages 57-81.

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