Rigorous Numerical Computation of Polynomial Differential Equations Over Unbounded Domains

  • Olivier Bournez
  • Daniel S. GraçaEmail author
  • Amaury Pouly
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9582)


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.



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.


  1. 1.
    Atkinson, K.E.: An Introduction to Numerical Analysis, 2nd edn. Wiley, New York (1989)zbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kawamura, A.: Lipschitz continuous ordinary differential equations are polynomial-space complete. Comput. Complex. 19(2), 305–332 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ko, K.I.: Computational Complexity of Real Functions. Birkhäuser, Basel (1991)CrossRefzbMATHGoogle 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). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

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

Personalised recommendations