IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v458y2023ics009630032300406x.html
   My bibliography  Save this article

Quadratized Taylor series methods for ODE numerical integration

Author

Listed:
  • Borri, Alessandro
  • Carravetta, Francesco
  • Palumbo, Pasquale

Abstract

We focus on Taylor Series Methods (TSM) and Automatic Differentiation (AD) for the numerical solution of Ordinary Differential Equations (ODE) characterized by a vector field given by a finite composition of elementary and standard functions. We show that computational advantages are achieved if a kind of pre-processing said Exact Quadratization (EQ) is applied to the ODE before applying the TSM and the AD. In particular, when the ODE function is given by a formal polynomial (i.e. with real powers) of n variables and m monomials, the computational complexity required by our EQ based method for the calculation of the k-th order Taylor coefficient is O(k) whereas by using the existing AD methods it amounts to O(k2).

Suggested Citation

  • Borri, Alessandro & Carravetta, Francesco & Palumbo, Pasquale, 2023. "Quadratized Taylor series methods for ODE numerical integration," Applied Mathematics and Computation, Elsevier, vol. 458(C).
  • Handle: RePEc:eee:apmaco:v:458:y:2023:i:c:s009630032300406x
    DOI: 10.1016/j.amc.2023.128237
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S009630032300406X
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.amc.2023.128237?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Amodio, P. & Iavernaro, F. & Mazzia, F. & Mukhametzhanov, M.S. & Sergeyev, Ya.D., 2017. "A generalized Taylor method of order three for the solution of initial value problems in standard and infinity floating-point arithmetic," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 141(C), pages 24-39.
    2. Abad, A. & Barrio, R. & Marco-Buzunariz, M. & Rodríguez, M., 2015. "Automatic implementation of the numerical Taylor series method: A Mathematica and Sage approach," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 227-245.
    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. Falcone, Alberto & Garro, Alfredo & Mukhametzhanov, Marat S. & Sergeyev, Yaroslav D., 2021. "A Simulink-based software solution using the Infinity Computer methodology for higher order differentiation," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    2. Renato De Leone & Giovanni Fasano & Yaroslav D. Sergeyev, 2018. "Planar methods and grossone for the Conjugate Gradient breakdown in nonlinear programming," Computational Optimization and Applications, Springer, vol. 71(1), pages 73-93, September.
    3. Caldarola, Fabio, 2018. "The Sierpinski curve viewed by numerical computations with infinities and infinitesimals," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 321-328.
    4. Cococcioni, Marco & Pappalardo, Massimo & Sergeyev, Yaroslav D., 2018. "Lexicographic multi-objective linear programming using grossone methodology: Theory and algorithm," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 298-311.
    5. Renato Leone & Giovanni Fasano & Massimo Roma & Yaroslav D. Sergeyev, 2020. "Iterative Grossone-Based Computation of Negative Curvature Directions in Large-Scale Optimization," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 554-589, August.
    6. Fiaschi, Lorenzo & Cococcioni, Marco, 2021. "Non-Archimedean game theory: A numerical approach," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    7. Essam R. El-Zahar & José Tenreiro Machado & Abdelhalim Ebaid, 2019. "A New Generalized Taylor-Like Explicit Method for Stiff Ordinary Differential Equations," Mathematics, MDPI, vol. 7(12), pages 1-18, December.

    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:eee:apmaco:v:458:y:2023:i:c:s009630032300406x. 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: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    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.