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Quadratized Taylor series methods for ODE numerical integration

Author

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  • 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
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    References listed on IDEAS

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    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.
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