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Purely Iterative Algorithms for Newton’s Maps and General Convergence

Author

Listed:
  • Sergio Amat

    (Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, 30202 Cartagena, Spain
    These authors contributed equally to this work.)

  • Rodrigo Castro

    (Facultad de Ciencias, Universidad de Valparaíso, Valparaíso 2340000, Chile
    These authors contributed equally to this work.)

  • Gerardo Honorato

    (CIMFAV and Institute of Mathematical Engineering, Universidad de Valparaíso, General Cruz 222, Valparaíso 2340000, Chile
    These authors contributed equally to this work.)

  • Á. A. Magreñán

    (Departamento de Matemáticas y Computación, Universidad de La Rioja, 26006 Logroño, Spain)

Abstract

The aim of this paper is to study the local dynamical behaviour of a broad class of purely iterative algorithms for Newton’s maps. In particular, we describe the nature and stability of fixed points and provide a type of scaling theorem. Based on those results, we apply a rigidity theorem in order to study the parameter space of cubic polynomials, for a large class of new root finding algorithms. Finally, we study the relations between critical points and the parameter space.

Suggested Citation

  • Sergio Amat & Rodrigo Castro & Gerardo Honorato & Á. A. Magreñán, 2020. "Purely Iterative Algorithms for Newton’s Maps and General Convergence," Mathematics, MDPI, vol. 8(7), pages 1-27, July.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:7:p:1158-:d:384712
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    References listed on IDEAS

    as
    1. Honorato, Gerardo & Plaza, Sergio, 2015. "Dynamical aspects of some convex acceleration methods as purely iterative algorithm for Newton’s maps," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 507-520.
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    Cited by:

    1. Uriel Filobello-Nino & Hector Vazquez-Leal & Jesús Huerta-Chua & Jaime Martínez-Castillo & Agustín L. Herrera-May & Mario Alberto Sandoval-Hernandez & Victor Manuel Jimenez-Fernandez, 2022. "The Enhanced Fixed Point Method: An Extremely Simple Procedure to Accelerate the Convergence of the Fixed Point Method to Solve Nonlinear Algebraic Equations," Mathematics, MDPI, vol. 10(20), pages 1-19, October.

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