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On the election of the damped parameter of a two-step relaxed Newton-type method

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Abstract

In this paper, we are interested to justified two typical hypotheses that appear in the convergence analysis, \(|\lambda |\le 2\) and \(z_0\) sufficient close to \(z^*\). In order to proof these ideas, the dynamics of a damped two-step Newton-type method for solving nonlinear equations and systems is presented. We present the parameter space for values of the damping factor in the complex plane, focusing our attention in such values for which the fixed points related to the roots are attracting. Moreover, we study the stability of the strange fixed points, showing that there exists attracting cycles and chaotical behavior for some choices of the damping factor.

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Acknowledgments

This research was supported by MICINN-FEDER MTM2010-17508 (Spain), by 08662/PI/08 (Murcia), by Ministerio de Ciencia y Tecnología MTM2014-52016-C2-1-P and by the Universidad Internacional de La Rioja (UNIR, http://www.unir.net), under the Plan Propio de Investigación, Desarrollo e Innovación [2013–2015]. Research group: Matemática aplicada al mundo real (MAMUR).

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Authors and Affiliations

  1. Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Cartagena, Spain

    Sergio Amat, Sonia Busquier & Concepción Bermúdez

  2. Escuela de Ingeniería, Universidad Internacional de La Rioja, Logroño, Spain

    Á. Alberto Magreñán

Authors
  1. Sergio Amat
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  2. Sonia Busquier
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  3. Concepción Bermúdez
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  4. Á. Alberto Magreñán
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Correspondence to Á. Alberto Magreñán.

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Amat, S., Busquier, S., Bermúdez, C. et al. On the election of the damped parameter of a two-step relaxed Newton-type method. Nonlinear Dyn 84, 9–18 (2016). https://doi.org/10.1007/s11071-015-2179-x

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  • DOI: https://doi.org/10.1007/s11071-015-2179-x

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