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A generalized Taylor method of order three for the solution of initial value problems in standard and infinity floating-point arithmetic

Author

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  • Amodio, P.
  • Iavernaro, F.
  • Mazzia, F.
  • Mukhametzhanov, M.S.
  • Sergeyev, Ya.D.

Abstract

A well-known drawback of algorithms based on Taylor series formulae is that the explicit calculation of higher order derivatives formally is an over-elaborate task. To avoid the analytical computation of the successive derivatives, numeric and automatic differentiation are usually used. A recent alternative to these techniques is based on the calculation of higher derivatives by using the Infinity Computer—a new computational device allowing one to work numerically with infinities and infinitesimals. Two variants of a one-step multi-point method closely related to the classical Taylor formula of order three are considered. It is shown that the new formula is order three accurate, though requiring only the first two derivatives of y(t) (rather than three if compared with the corresponding Taylor formula of order three). To get numerical evidence of the theoretical results, a few test problems are solved by means of the new methods and the obtained results are compared with the performance of Taylor methods of order up to four.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:141:y:2017:i:c:p:24-39
    DOI: 10.1016/j.matcom.2016.03.007
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    References listed on IDEAS

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    1. Kanovei, Vladimir & Lyubetsky, Vassily, 2015. "Grossone approach to Hutton and Euler transforms," Applied Mathematics and Computation, Elsevier, vol. 255(C), pages 36-43.
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    4. Sergeyev, Yaroslav D., 2007. "Blinking fractals and their quantitative analysis using infinite and infinitesimal numbers," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 50-75.
    5. Lolli, Gabriele, 2015. "Metamathematical investigations on the theory of Grossone," Applied Mathematics and Computation, Elsevier, vol. 255(C), pages 3-14.
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    Cited by:

    1. 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.
    2. Fiaschi, Lorenzo & Cococcioni, Marco, 2021. "Non-Archimedean game theory: A numerical approach," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    3. 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.
    4. 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.
    5. Borri, Alessandro & Carravetta, Francesco & Palumbo, Pasquale, 2023. "Quadratized Taylor series methods for ODE numerical integration," Applied Mathematics and Computation, Elsevier, vol. 458(C).
    6. 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.
    7. 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).
    8. 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.

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