IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i11p1219-d563413.html
   My bibliography  Save this article

Advances in the Approximation of the Matrix Hyperbolic Tangent

Author

Listed:
  • Javier Ibáñez

    (Instituto de Instrumentación para Imagen Molecular, Universitat Politècnica de València, Av. dels Tarongers, 14, 46011 Valencia, Spain)

  • José M. Alonso

    (Instituto de Instrumentación para Imagen Molecular, Universitat Politècnica de València, Av. dels Tarongers, 14, 46011 Valencia, Spain)

  • Jorge Sastre

    (Instituto de Telecomunicación y Aplicaciones Multimedia, Universitat Politècnica de València, Ed. 8G, Camino de Vera s/n, 46022 Valencia, Spain)

  • Emilio Defez

    (Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Ed. 8G, Camino de Vera s/n, 46022 Valencia, Spain)

  • Pedro Alonso-Jordá

    (Department of Computer Systems and Computation, Universitat Politècnica de València, Ed. 1F, Camino de Vera s/n, 46022 Valencia, Spain)

Abstract

In this paper, we introduce two approaches to compute the matrix hyperbolic tangent. While one of them is based on its own definition and uses the matrix exponential, the other one is focused on the expansion of its Taylor series. For this second approximation, we analyse two different alternatives to evaluate the corresponding matrix polynomials. This resulted in three stable and accurate codes, which we implemented in MATLAB and numerically and computationally compared by means of a battery of tests composed of distinct state-of-the-art matrices. Our results show that the Taylor series-based methods were more accurate, although somewhat more computationally expensive, compared with the approach based on the exponential matrix. To avoid this drawback, we propose the use of a set of formulas that allows us to evaluate polynomials in a more efficient way compared with that of the traditional Paterson–Stockmeyer method, thus, substantially reducing the number of matrix products (practically equal in number to the approach based on the matrix exponential), without penalising the accuracy of the result.

Suggested Citation

  • Javier Ibáñez & José M. Alonso & Jorge Sastre & Emilio Defez & Pedro Alonso-Jordá, 2021. "Advances in the Approximation of the Matrix Hyperbolic Tangent," Mathematics, MDPI, vol. 9(11), pages 1-20, May.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:11:p:1219-:d:563413
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/11/1219/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/11/1219/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Sastre, J. & Ibáñez, J. & Defez, E., 2019. "Boosting the computation of the matrix exponential," Applied Mathematics and Computation, Elsevier, vol. 340(C), pages 206-220.
    2. Constantine, A. G. & Muirhead, R. J., 1972. "Partial differential equations for hypergeometric functions of two argument matrices," Journal of Multivariate Analysis, Elsevier, vol. 2(3), pages 332-338, September.
    Full references (including those not matched with items on IDEAS)

    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. Yasuko Chikuse, 1976. "Partial differential equations for hypergeometric functions of complex argument matrices and their applications," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 28(1), pages 187-199, December.
    2. Javier Ibáñez & Jorge Sastre & Pedro Ruiz & José M. Alonso & Emilio Defez, 2021. "An Improved Taylor Algorithm for Computing the Matrix Logarithm," Mathematics, MDPI, vol. 9(17), pages 1-19, August.
    3. Jorge Sastre & Javier Ibáñez, 2021. "Efficient Evaluation of Matrix Polynomials beyond the Paterson–Stockmeyer Method," Mathematics, MDPI, vol. 9(14), pages 1-23, July.
    4. Ghazi S. Khammash & Praveen Agarwal & Junesang Choi, 2020. "Extended k-Gamma and k-Beta Functions of Matrix Arguments," Mathematics, MDPI, vol. 8(10), pages 1-13, October.
    5. Bader, Philipp & Blanes, Sergio & Casas, Fernando & Seydaoğlu, Muaz, 2022. "An efficient algorithm to compute the exponential of skew-Hermitian matrices for the time integration of the Schrödinger equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 383-400.
    6. Ahmed Bakhet & Fuli He, 2020. "On 2-Variables Konhauser Matrix Polynomials and Their Fractional Integrals," Mathematics, MDPI, vol. 8(2), pages 1-12, February.

    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:9:y:2021:i:11:p:1219-:d:563413. 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.