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An implicit algorithm for validated enclosures of the solutions to variational equations for ODEs

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  • Walawska, Irmina
  • Wilczak, Daniel

Abstract

We propose a new algorithm for computing validated bounds for the solutions to the first order variational equations associated to ODEs. These validated solutions are the kernel of numerics computer-assisted proofs in dynamical systems literature. The method uses a high-order Taylor method as a predictor step and an implicit method based on the Hermite–Obreshkov interpolation as a corrector step. The proposed algorithm is an improvement of the C1-Lohner algorithm proposed by Zgliczyński and it provides sharper bounds.

Suggested Citation

  • Walawska, Irmina & Wilczak, Daniel, 2016. "An implicit algorithm for validated enclosures of the solutions to variational equations for ODEs," Applied Mathematics and Computation, Elsevier, vol. 291(C), pages 303-322.
    • Handle: RePEc:eee:apmaco:v:291:y:2016:i:c:p:303-322
    DOI: 10.1016/j.amc.2016.07.005
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    1. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(3), pages 427-432, June.
    2. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(4), pages 629-637, August.
    3. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(5), pages 777-788, October.
    4. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(1), pages 151-160, February.
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    Cited by:

    1. Le Mézo, Thomas & Jaulin, Luc & Zerr, Benoît, 2018. "Bracketing the solutions of an ordinary differential equation with uncertain initial conditions," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 70-79.

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