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Nonlinear Dynamics

, Volume 85, Issue 3, pp 1939–1954 | Cite as

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

  • Alicia CorderoEmail author
  • Antonio Franques
  • Juan R. Torregrosa
Original Paper

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.

Keywords

Nonlinear equations Iterative methods Dynamical behavior Parameter plane Convergence regions Bratu problem 

Notes

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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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Alicia Cordero
    • 1
    Email author
  • Antonio Franques
    • 1
  • Juan R. Torregrosa
    • 1
  1. 1.Instituto Universitario de Matemática MultidisciplinarUniversitat Politècnica de ValènciaValenciaSpain

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