Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains

  • Olivier Bournez
  • Daniel S. Graça
  • Amaury Pouly
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6907)


In this paper we consider the computational complexity of solving initial-value problems defined with analytic ordinary differential equations (ODEs) over unbounded domains of ℝ n and ℂ n , under the Computable Analysis setting. We show that the solution can be computed in polynomial time over its maximal interval of definition, provided it satisfies a very generous bound on its growth, and that the function admits an analytic extension to the complex plane.


Polynomial Time Taylor Series Turing Machine Lipschitz Condition Unbounded Domain 
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  1. 1.
    Ko, K.I.: Computational Complexity of Real Functions. Birkhäuser, Basel (1991)Google Scholar
  2. 2.
    Shannon, C.E.: Mathematical theory of the differential analyzer. J. Math. Phys. 20, 337–354 (1941)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bush, V.: The differential analyzer. A new machine for solving differential equations. J. Franklin Inst. 212, 447–488 (1931)Google Scholar
  4. 4.
    Graça, D.S., Costa, J.F.: Analog computers and recursive functions over the reals. J. Complexity 19(5), 644–664 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Graça, D., Zhong, N., Buescu, J.: Computability, noncomputability and undecidability of maximal intervals of IVPs. Trans. Amer. Math. Soc. 361(6), 2913–2927 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Collins, P., Graça, D.S.: Effective computability of solutions of differential inclusions — the ten thousand monkeys approach. Journal of Universal Computer Science 15(6), 1162–1185 (2009)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Pour-El, M.B., Richards, J.I.: A computable ordinary differential equation which possesses no computable solution. Ann. Math. Logic 17, 61–90 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Demailly, J.-P.: Analyse Numérique et Equations Différentielles. Presses Universitaires de Grenoble (1991)Google Scholar
  9. 9.
    Smith, W.D.: Church’s thesis meets the N-body problem. Applied Mathematics and Computation 178(1), 154–183 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Perko, L.: Differential Equations and Dynamical Systems, 3rd edn. Springer, Heidelberg (2001)zbMATHGoogle Scholar
  11. 11.
    Ruohonen, K.: An effective Cauchy-Peano existence theorem for unique solutions. Internat. J. Found. Comput. Sci. 7(2), 151–160 (1996)zbMATHCrossRefGoogle Scholar
  12. 12.
    Kawamura, A.: Lipschitz continuous ordinary differential equations are polynomial-space complete. In: 2009 24th Annual IEEE Conference on Computational Complexity, pp. 149–160. IEEE, Los Alamitos (2009)CrossRefGoogle Scholar
  13. 13.
    Müller, N., Moiske, B.: Solving initial value problems in polynomial time. In: Proc. 22 JAIIO - PANEL 1993, Part 2, pp. 283–293 (1993)Google Scholar
  14. 14.
    Müller, N.T., Korovina, M.V.: Making big steps in trajectories. Electr. Proc. Theoret. Comput. Sci. 24, 106–119 (2010)CrossRefGoogle Scholar
  15. 15.
    Birkhoff, G., Rota, G.C.: Ordinary Differential Equations, 4th edn. John Wiley, Chichester (1989)Google Scholar
  16. 16.
    Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)zbMATHGoogle Scholar
  17. 17.
    Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proc. London Math. Soc. (Ser.2–42), 230–265 (1936)Google Scholar
  18. 18.
    Grzegorczyk, A.: On the definitions of computable real continuous functions. Fund. Math. 44, 61–71 (1957)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Lacombe, D.: Extension de la notion de fonction récursive aux fonctions d’une ou plusieurs variables réelles III. C. R. Acad. Sci. Paris 241, 151–153 (1955)Google Scholar
  20. 20.
    Weihrauch, K.: Computable Analysis: an Introduction. Springer, Heidelberg (2000)zbMATHGoogle Scholar
  21. 21.
    Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Springer, Heidelberg (1989)zbMATHGoogle Scholar
  22. 22.
    Ko, K.I., Friedman, H.: Computational complexity of real functions. Theoret. Comput. Sci. 20, 323–352 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Müller, N.T.: Uniform computational complexity of taylor series. In: Ottmann, T. (ed.) ICALP 1987. LNCS, vol. 267, pp. 435–444. Springer, Heidelberg (1987)Google Scholar
  24. 24.
    Graça, D.S., Campagnolo, M.L., Buescu, J.: Computability with polynomial differential equations. Adv. Appl. Math. 40(3), 330–349 (2008)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2011

Authors and Affiliations

  • Olivier Bournez
    • 1
  • Daniel S. Graça
    • 2
    • 3
  • Amaury Pouly
    • 4
  1. 1.Ecole Polytechnique, LIXPalaiseau CedexFrance
  2. 2.CEDMES/FCTUniversidade do Algarve, C. GambelasFaroPortugal
  3. 3.SQIG /Instituto de TelecomunicaçõesLisbonPortugal
  4. 4.Ecole Normale Supérieure de LyonFrance

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