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The Enhanced Fixed Point Method: An Extremely Simple Procedure to Accelerate the Convergence of the Fixed Point Method to Solve Nonlinear Algebraic Equations

Author

Listed:
  • Uriel Filobello-Nino

    (Facultad de Instrumentación Electrónica, Universidad Veracruzana, Cto. Gonzalo Aguirre Beltrán S/N, Xalapa 91000, Veracruz, Mexico)

  • Hector Vazquez-Leal

    (Facultad de Instrumentación Electrónica, Universidad Veracruzana, Cto. Gonzalo Aguirre Beltrán S/N, Xalapa 91000, Veracruz, Mexico
    Consejo Veracruzano de Investigación Científica y Desarrollo Tecnológico (COVEICYDET), Av Rafael Murillo Vidal No. 1735, Cuauhtémoc, Xalapa 91069, Veracruz, Mexico)

  • Jesús Huerta-Chua

    (Instituto Tecnológico Superior de Poza Rica, Tecnológico Nacional de México, Luis Donaldo Colosio Murrieta S/N, Arroyo del Maíz, Poza Rica 93230, Veracruz, Mexico)

  • Jaime Martínez-Castillo

    (Centro de Investigación en Micro y Nanotecnología, Universidad Veracruzana, Boca del Río 94294, Veracruz, Mexico)

  • Agustín L. Herrera-May

    (Centro de Investigación en Micro y Nanotecnología, Universidad Veracruzana, Boca del Río 94294, Veracruz, Mexico
    Facultad de Ingeniería de la Construcción y el Hábitat, Universidad Veracruzana, Boca del Río 94294, Veracruz, Mexico)

  • Mario Alberto Sandoval-Hernandez

    (Instituto Tecnológico Superior de Poza Rica, Tecnológico Nacional de México, Luis Donaldo Colosio Murrieta S/N, Arroyo del Maíz, Poza Rica 93230, Veracruz, Mexico)

  • Victor Manuel Jimenez-Fernandez

    (Facultad de Instrumentación Electrónica, Universidad Veracruzana, Cto. Gonzalo Aguirre Beltrán S/N, Xalapa 91000, Veracruz, Mexico)

Abstract

This work proposes the Enhanced Fixed Point Method (EFPM) as a straightforward modification to the problem of finding an exact or approximate solution for a linear or nonlinear algebraic equation. The proposal consists of providing a versatile method that is easy to employ and systematic. Therefore, it is expected that this work contributes to breaking the paradigm that an effective modification for a known method has to be necessarily long and complicated. As a matter of fact, the method expresses an algebraic equation in terms of the same equation but multiplied for an adequate factor, which most of the times is just a simple numeric factor. The main idea is modifying the original equation, slightly changing it for others in such a way that both have the same solution. Next, the modified equation is expressed as a fixed point problem and the proposed parameters are employed to accelerate the convergence of the fixed point problem for the original equation. Since the Newton method results from a possible fixed point problem of an algebraic equation, we will see that it is relatively easy to get modified versions of the Newton method with orders of convergence major than two. We will see in this work the convenience of this procedure.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:20:p:3797-:d:942787
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    References listed on IDEAS

    as
    1. 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.
    2. Mohammed Barrada & Mariya Ouaissa & Yassine Rhazali & Mariyam Ouaissa, 2020. "A New Class of Halley’s Method with Third-Order Convergence for Solving Nonlinear Equations," Journal of Applied Mathematics, Hindawi, vol. 2020, pages 1-13, July.
    3. Tahereh Eftekhari, 2014. "A New Sixth-Order Steffensen-Type Iterative Method for Solving Nonlinear Equations," International Journal of Analysis, Hindawi, vol. 2014, pages 1-5, February.
    4. J. P. Jaiswal, 2015. "Two Bi-Accelerator Improved with Memory Schemes for Solving Nonlinear Equations," Discrete Dynamics in Nature and Society, Hindawi, vol. 2015, pages 1-7, February.
    5. Beny Neta, 2021. "A New Derivative-Free Method to Solve Nonlinear Equations," Mathematics, MDPI, vol. 9(6), pages 1-5, March.
    6. Xiaofeng Wang & Mingming Zhu, 2020. "Two Iterative Methods with Memory Constructed by the Method of Inverse Interpolation and Their Dynamics," Mathematics, MDPI, vol. 8(7), pages 1-12, July.
    7. Malik Zaka Ullah & A. S. Al-Fhaid & Fayyaz Ahmad, 2013. "Four-Point Optimal Sixteenth-Order Iterative Method for Solving Nonlinear Equations," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-5, September.
    8. Francisco I. Chicharro & Rafael A. Contreras & Neus Garrido, 2020. "A Family of Multiple-Root Finding Iterative Methods Based on Weight Functions," Mathematics, MDPI, vol. 8(12), pages 1-17, December.
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