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An optimal three-point eighth-order iterative method without memory for solving nonlinear equations with its dynamics

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Abstract

We present a three-point iterative method without memory for solving nonlinear equations in one variable. The proposed method provides convergence order eight with four function evaluations per iteration. Hence, it possesses a very high computational efficiency and supports Kung–Traub’s conjecture. The construction, the convergence analysis, and the numerical implementation of the method will be presented. Using several test problems, the proposed method will be compared with existing methods of convergence order eight concerning accuracy and basins of attraction. Furthermore, some measures are used to judge methods with respect to their performance in finding the basins of attraction.

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Acknowledgements

The research of the fourth author is supported by Grant MTM2015-65888-C4-4-P (MINECO/FEDER).

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Authors and Affiliations

  1. Institut für Numerische Mathematik, Technische Universität Dresden, 01062, Dresden, Germany

    Gunar Matthies

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

    Mehdi Salimi

  3. Department of Mathematics, Universiti Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia

    Mehdi Salimi

  4. MEDAlics, Research Center at Università per Stranieri Dante Alighieri, Reggio Calabria, Italy

    Somayeh Sharifi

  5. Departamento de Matemáticas y Computación, Universidad de La Rioja, Logroño, Spain

    Juan Luis Varona

Authors
  1. Gunar Matthies
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  2. Mehdi Salimi
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  3. Somayeh Sharifi
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  4. Juan Luis Varona
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Correspondence to Mehdi Salimi.

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Matthies, G., Salimi, M., Sharifi, S. et al. An optimal three-point eighth-order iterative method without memory for solving nonlinear equations with its dynamics. Japan J. Indust. Appl. Math. 33, 751–766 (2016). https://doi.org/10.1007/s13160-016-0229-5

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  • DOI: https://doi.org/10.1007/s13160-016-0229-5

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