IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i2p274-d322182.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/2/274/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/8/2/274/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. F. I. Chicharro & A. Cordero & J. R. Torregrosa & M. P. Vassileva, 2017. "King-Type Derivative-Free Iterative Families: Real and Memory Dynamics," Complexity, Hindawi, vol. 2017, pages 1-15, October.
    2. Geum, Young Hee & Kim, Young Ik & Neta, Beny, 2016. "A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points," Applied Mathematics and Computation, Elsevier, vol. 283(C), pages 120-140.
    3. Geum, Young Hee & Kim, Young Ik & Neta, Beny, 2015. "On developing a higher-order family of double-Newton methods with a bivariate weighting function," Applied Mathematics and Computation, Elsevier, vol. 254(C), pages 277-290.
    4. Alicia Cordero & Javier G. Maimó & Juan R. Torregrosa & María P. Vassileva, 2023. "Improving Newton–Schulz Method for Approximating Matrix Generalized Inverse by Using Schemes with Memory," Mathematics, MDPI, vol. 11(14), pages 1-19, July.
    5. Alicia Cordero & Javier G. Maimó & Juan R. Torregrosa & María P. Vassileva, 2019. "Iterative Methods with Memory for Solving Systems of Nonlinear Equations Using a Second Order Approximation," Mathematics, MDPI, vol. 7(11), pages 1-12, November.
    6. Chun, Changbum & Neta, Beny, 2016. "An analysis of a Khattri’s 4th order family of methods," Applied Mathematics and Computation, Elsevier, vol. 279(C), pages 198-207.
    7. Geum, Young Hee & Kim, Young Ik & Magreñán, Á. Alberto, 2016. "A biparametric extension of King’s fourth-order methods and their dynamics," Applied Mathematics and Computation, Elsevier, vol. 282(C), pages 254-275.
    8. Geum, Young Hee & Kim, Young Ik & Neta, Beny, 2015. "A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 387-400.
    9. Alicia Cordero & Beny Neta & Juan R. Torregrosa, 2021. "Memorizing Schröder’s Method as an Efficient Strategy for Estimating Roots of Unknown Multiplicity," Mathematics, MDPI, vol. 9(20), pages 1-13, October.
    10. Cordero, Alicia & Soleymani, Fazlollah & Torregrosa, Juan R. & Haghani, F. Khaksar, 2017. "A family of Kurchatov-type methods and its stability," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 264-279.
    11. Xiaofeng Wang & Qiannan Fan, 2020. "A Modified Ren’s Method with Memory Using a Simple Self-Accelerating Parameter," Mathematics, MDPI, vol. 8(4), pages 1-12, April.
    12. Geum, Young Hee & Kim, Young Ik & Neta, Beny, 2017. "A family of optimal quartic-order multiple-zero finders with a weight function of the principal kth root of a derivative-to-derivative ratio and their basins of attraction," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 136(C), pages 1-21.
    13. 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.
    14. Lee, Min-Young & Ik Kim, Young & Alberto Magreñán, Á., 2017. "On the dynamics of a triparametric family of optimal fourth-order multiple-zero finders with a weight function of the principal mth root of a function-to function ratio," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 564-590.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:8:y:2020:i:2:p:274-:d:322182. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.