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Stability analysis of a parametric family of iterative methods for solving nonlinear models

Author

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  • Cordero, Alicia
  • Gutiérrez, José M.
  • Magreñán, Á. Alberto
  • Torregrosa, Juan R.

Abstract

A one-parametric family of fourth-order iterative methods for solving nonlinear systems is presented, proving the fourth-order of convergence of all members in this family, except one of them whose order is five. The methods in our family are numerically compared with other known methods in terms of the classical efficiency index (order of convergence and number of functional evaluations) and in terms of the operational efficiency index, which also takes into account the total number of product-quotients per iteration. In order to analyze its stability and its dynamical properties, the parameter space for quadratic polynomials is shown. The stability of the strange fixed points is studied in this case. We note that even for this particular case, the family presents a very interesting dynamical behavior. The analysis of the parameter plane allows us to find values for the involved parameter with good stability properties as well as other values with bad numerical behavior. Finally, amongst the first ones, there is a special value of the parameter related to a fifth-order method in the family.

Suggested Citation

  • Cordero, Alicia & Gutiérrez, José M. & Magreñán, Á. Alberto & Torregrosa, Juan R., 2016. "Stability analysis of a parametric family of iterative methods for solving nonlinear models," Applied Mathematics and Computation, Elsevier, vol. 285(C), pages 26-40.
    • Handle: RePEc:eee:apmaco:v:285:y:2016:i:c:p:26-40
    DOI: 10.1016/j.amc.2016.03.021
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    Cited by:

    1. Amiri, Abdolreza & Cordero, Alicia & Taghi Darvishi, M. & Torregrosa, Juan R., 2018. "Stability analysis of a parametric family of seventh-order iterative methods for solving nonlinear systems," Applied Mathematics and Computation, Elsevier, vol. 323(C), pages 43-57.
    2. Ramandeep Behl & Ioannis K. Argyros & Fouad Othman Mallawi, 2021. "Some High-Order Convergent Iterative Procedures for Nonlinear Systems with Local Convergence," Mathematics, MDPI, vol. 9(12), pages 1-13, June.
    3. Campos, B. & Vindel, P., 2021. "Dynamics of subfamilies of Ostrowski–Chun methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 181(C), pages 57-81.
    4. Cristina Amorós & Ioannis K. Argyros & Á. Alberto Magreñán & Samundra Regmi & Rubén González & Juan Antonio Sicilia, 2019. "Extending the Applicability of Stirling’s Method," Mathematics, MDPI, vol. 8(1), pages 1-10, December.
    5. Cristina Amorós & Ioannis K. Argyros & Daniel González & Ángel Alberto Magreñán & Samundra Regmi & Íñigo Sarría, 2020. "New Improvement of the Domain of Parameters for Newton’s Method," Mathematics, MDPI, vol. 8(1), pages 1-12, January.

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