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Stability analysis of a family of optimal fourth-order methods for multiple roots

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Abstract

Complex dynamics tools applied on the rational functions resulting from a parametric family of roots solvers for nonlinear equations provide very useful results that have been stated in the last years. These qualitative properties allow the user to select the most efficient members from the family of iterative schemes, in terms of stability and wideness of the sets of convergent initial guesses. These tools have been widely used in the case of iterative procedures for finding simple roots and only recently are being applied on the case of multiplicity m > 1. In this paper, by using weight function procedure, we design a general class of iterative methods for calculating multiple roots that includes some known methods. In this class, conditions on the weight function are not very restrictive, so a large number of different subfamilies can be generated, all of them are optimal with fourth-order of convergence. Their dynamical analysis gives us enough information to select those with better properties and test them on different numerical experiments, showing their numerical properties.

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Acknowledgments

The authors would like to thank the anonymous reviewers for their help to improve the final version of this manuscript.

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Authors and Affiliations

  1. Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, 60800, Pakistan

    Fiza Zafar

  2. Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, 46022, València, Spain

    Fiza Zafar, Alicia Cordero & Juan R. Torregrosa

Authors
  1. Fiza Zafar
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  2. Alicia Cordero
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  3. Juan R. Torregrosa
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Corresponding author

Correspondence to Juan R. Torregrosa.

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The authors declare that there are no conflicts of interest.

Additional information

This research was partially supported by Ministerio de Economía y Competitividad MTM2014-52016-C2-2-P, Generalitat Valenciana PROMETEO/2016/089 and Schlumberger Foundation-Faculty for Future Program.

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Zafar, F., Cordero, A. & Torregrosa, J.R. Stability analysis of a family of optimal fourth-order methods for multiple roots. Numer Algor 81, 947–981 (2019). https://doi.org/10.1007/s11075-018-0577-0

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  • DOI: https://doi.org/10.1007/s11075-018-0577-0

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