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A new class of three-point methods with optimal convergence order eight and its dynamics

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Abstract

We establish a new class of three-point methods for the computation of simple zeros of a scalar function. Based on the two-point optimal method by Ostrowski (1966), we construct a family of order eight methods which use three evaluations of f and one of f′ and therefore have an efficiency index equal to \(\sqrt [4]{8}\approx 1.682\) and are optimal in the sense of the Kung and Traub conjecture (Kung and Traub J. Assoc. Comput. Math. 21, 634–651, 1974). Moreover, the dynamics of the proposed methods are shown with some comparisons to other existing methods. Numerical comparison with existing optimal schemes suggests that the new class provides a valuable alternative for solving nonlinear equations.

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Authors and Affiliations

  1. Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran

    Taher Lotfi & Somayeh Sharifi

  2. Center for Dynamics, Department of Mathematics, Technische Universität Dresden, 01062, Dresden, Germany

    Mehdi Salimi & Stefan Siegmund

Authors
  1. Taher Lotfi
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  2. Somayeh Sharifi
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  3. Mehdi Salimi
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  4. Stefan Siegmund
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Correspondence to Stefan Siegmund.

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Lotfi, T., Sharifi, S., Salimi, M. et al. A new class of three-point methods with optimal convergence order eight and its dynamics. Numer Algor 68, 261–288 (2015). https://doi.org/10.1007/s11075-014-9843-y

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  • DOI: https://doi.org/10.1007/s11075-014-9843-y

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