Rigorous Numerical Computation of Polynomial Differential Equations Over Unbounded Domains

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9582)

Abstract

In this abstract we present a rigorous numerical algorithm which solves initial-value problems (IVPs) defined with polynomial differential equations (i.e. IVPs of the type \(y'=p(t,y)\), \(y(t_0)=y_0\), where p is a vector of polynomials) for any value of t. The inputs of the algorithm are the data defining the initial-value problem, the time T at which we want to compute the solution of the IVP, and the maximum allowable error \(\varepsilon >0\). Using these inputs, the algorithm will output a value \(\tilde{y}_T\) such that \(\Vert \tilde{y}_T-y(T)\Vert \le \varepsilon \) in time polynomial in T, \(-\log \varepsilon \), and in several quantities related to the polynomial IVP.

Notes

Acknowledgments

D. Graça was partially supported by Fundação para a Ciência e a Tecnologia and EU FEDER POCTI/POCI via SQIG - Instituto de Telecomunicações through the FCT project UID/EEA/50008/2013.

References

  1. 1.
    Atkinson, K.E.: An Introduction to Numerical Analysis, 2nd edn. Wiley, New York (1989)MATHGoogle Scholar
  2. 2.
    Bostan, A., Chyzak, F., Ollivier, F., Salvy, B., Schost, É., Sedoglavic, A.: Fast computation of power series solutions of systems of differential equations. In: SODA 2007, pp. 1012–1021, January 2007Google Scholar
  3. 3.
    Graça, D.S., Campagnolo, M.L., Buescu, J.: Computability with polynomial differential equations. Adv. Appl. Math. 40(3), 330–349 (2008)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Kawamura, A.: Lipschitz continuous ordinary differential equations are polynomial-space complete. Comput. Complex. 19(2), 305–332 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ko, K.I.: Computational Complexity of Real Functions. Birkhäuser, Basel (1991)CrossRefMATHGoogle Scholar
  6. 6.
    Lorenz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141 (1963)CrossRefGoogle Scholar
  7. 7.
    Müller, N., Moiske, B.: Solving initial value problems in polynomial time. In: Proceedings of 22 JAIIO - PANEL 1993, Part 2, pp. 283–293 (1993)Google Scholar
  8. 8.
    Pouly, A.: Continuous Models of Computation: From Computability to Complexity. Ph.D. thesis, Ecole Polytechnique/Universidade do Algarve (2015)Google Scholar
  9. 9.
    Warne, P.G., Warne, D.P., Sochacki, J.S., Parker, G.E., Carothers, D.C.: Explicit a-priori error bounds and adaptive error control for approximation of nonlinear initial value differential systems. Comput. Math. Appl. 52(12), 1695–1710 (2006). http://dx.doi.org/10.1016/j.camwa.2005.12.004 MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Olivier Bournez
    • 1
  • Daniel S. Graça
    • 2
    • 3
  • Amaury Pouly
    • 1
    • 2
  1. 1.LIXEcole PolytechniquePalaiseau CedexFrance
  2. 2.CEDMES/FCTUniversidade do AlgarveFaroPortugal
  3. 3.SQIG/Instituto de TelecomunicaçõesLisbonPortugal

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