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Chaos and convergence of a family generalizing Homeier’s method with damping parameters

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Abstract

In this paper, a family of parametric iterative methods for solving nonlinear equations, including Homeier’s scheme, is presented. Its local convergence is obtained and the dynamical behavior on quadratic polynomials of the resulting family is studied in order to choose those values of the parameter that ensure stable behavior. To get this aim, the analysis of fixed and critical points and the associated parameter plane show the dynamical richness of the family and allow us to find members of this class with good numerical properties and also other ones with pathological conduct. To check the stable behavior of the good selected ones, the discretized planar 1D-Bratu problem is solved. Some of those chosen members of the family achieve good results when Homeier’s scheme fails.

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Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable suggestions and comments that have substantially improved the final version of this manuscript.

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Authors and Affiliations

  1. Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera s/n, 46022, Valencia, Spain

    Alicia Cordero, Antonio Franques & Juan R. Torregrosa

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  1. Alicia Cordero
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  2. Antonio Franques
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  3. Juan R. Torregrosa
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Correspondence to Alicia Cordero.

Additional information

This research was supported by Ministerio de Economía y Competitividad MTM2014-52016-C02-2-P.

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Cordero, A., Franques, A. & Torregrosa, J.R. Chaos and convergence of a family generalizing Homeier’s method with damping parameters. Nonlinear Dyn 85, 1939–1954 (2016). https://doi.org/10.1007/s11071-016-2807-0

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