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Impact on Stability by the Use of Memory in Traub-Type Schemes

Author

Listed:
  • Francisco I. Chicharro

    (Escuela Superior de Ingeniería y Tecnología, Universidad Internacional de La Rioja, 26006 Logroño, Spain)

  • Alicia Cordero

    (Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, 46022 València, Spain)

  • Neus Garrido

    (Escuela Superior de Ingeniería y Tecnología, Universidad Internacional de La Rioja, 26006 Logroño, Spain)

  • Juan R. Torregrosa

    (Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, 46022 València, Spain)

Abstract

In this work, two Traub-type methods with memory are introduced using accelerating parameters. To obtain schemes with memory, after the inclusion of these parameters in Traub’s method, they have been designed using linear approximations or the Newton’s interpolation polynomials. In both cases, the parameters use information from the current and the previous iterations, so they define a method with memory. Moreover, they achieve higher order of convergence than Traub’s scheme without any additional functional evaluations. The real dynamical analysis verifies that the proposed methods with memory not only converge faster, but they are also more stable than the original scheme. The methods selected by means of this analysis can be applied for solving nonlinear problems with a wider set of initial estimations than their original partners. This fact also involves a lower number of iterations in the process.

Suggested Citation

  • Francisco I. Chicharro & Alicia Cordero & Neus Garrido & Juan R. Torregrosa, 2020. "Impact on Stability by the Use of Memory in Traub-Type Schemes," Mathematics, MDPI, vol. 8(2), pages 1-16, February.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:2:p:274-:d:322182
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    References listed on IDEAS

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    1. Campos, Beatriz & Cordero, Alicia & Torregrosa, Juan R. & Vindel, Pura, 2015. "A multidimensional dynamical approach to iterative methods with memory," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 701-715.
    2. Magreñán, Á. Alberto & Cordero, Alicia & Gutiérrez, José M. & Torregrosa, Juan R., 2014. "Real qualitative behavior of a fourth-order family of iterative methods by using the convergence plane," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 105(C), pages 49-61.
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    Cited by:

    1. Ramandeep Behl & Alicia Cordero & Juan R. Torregrosa & Sonia Bhalla, 2021. "A New High-Order Jacobian-Free Iterative Method with Memory for Solving Nonlinear Systems," Mathematics, MDPI, vol. 9(17), pages 1-16, September.

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