Nonlinear Dynamics

, Volume 87, Issue 2, pp 913–938 | Cite as

Widening basins of attraction of optimal iterative methods

  • Parisa Bakhtiari
  • Alicia Cordero
  • Taher Lotfi
  • Kathayoun Mahdiani
  • Juan R. TorregrosaEmail author
Original Paper


In this work, we analyze the dynamical behavior on quadratic polynomials of a class of derivative-free optimal parametric iterative methods, designed by Khattri and Steihaug. By using their parameter as an accelerator, we develop different methods with memory of orders three, six and twelve, without adding new functional evaluations. Then a dynamical approach is made, comparing each of the proposed methods with the original ones without memory, with the following empiric conclusion: Basins of attraction of iterative schemes with memory are wider and the behavior is more stable. This has been numerically checked by estimating the solution of a practical problem, as the friction factor of a pipe and also of other nonlinear academic problems.


Multi-point iterative methods Dynamical plane Basin of attraction With and without memory methods Kung and Traub’s conjecture Efficiency index 



The authors thank to the anonymous referees for their valuable comments and for the suggestions that have improved the final version of the paper.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Young Researchers and Elite Club, Hamedan BranchIslamic Azad UniversityHamedanIran
  2. 2.Instituto Universitario de Matemática MultidisciplinarUniversitat Politècnica de ValènciaValenciaSpain
  3. 3.Department of Mathematics, Hamedan BranchIslamic Azad UniversityHamedanIran

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