Computational Complexity of Smooth Differential Equations

  • Akitoshi Kawamura
  • Hiroyuki Ota
  • Carsten Rösnick
  • Martin Ziegler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)


The computational complexity of the solution h to the ordinary differential equation h(0) = 0, h′(t) = g(t, h(t)) under various assumptions on the function g has been investigated in hope of understanding the intrinsic hardness of solving the equation numerically. Kawamura showed in 2010 that the solution h can be PSPACE-hard even if g is assumed to be Lipschitz continuous and polynomial-time computable. We place further requirements on the smoothness of g and obtain the following results: the solution h can still be PSPACE-hard if g is assumed to be of class C 1; for each k ≥ 2, the solution h can be hard for the counting hierarchy if g is of class C k.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ko, K.: Complexity Theory of Real Functions. Birkhäuser Boston (1991)Google Scholar
  2. 2.
    Weihrauch, K.: Computable Analysis: An Introduction. Springer (2000)Google Scholar
  3. 3.
    Kawamura, A.: Lipschitz continuous ordinary differential equations are polynomial-space complete. Computational Complexity 19(2), 305–332 (2010)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Pour-El, M., Richards, I.: A computable ordinary differential equation which possesses no computable solution. Annals of Mathematical Logic 17(1-2), 61–90 (1979)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Coddington, E., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill (1955)Google Scholar
  6. 6.
    Ko, K.: On the computational complexity of ordinary differential equations. Information and Control 58(1-3), 157–194 (1983)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Miller, W.: Recursive function theory and numerical analysis. Journal of Computer and System Sciences 4(5), 465–472 (1970)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Müller, N.: Uniform Computational Complexity of Taylor Series. In: Ottmann, T. (ed.) ICALP 1987. LNCS, vol. 267, pp. 435–444. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  9. 9.
    Ko, K., Friedman, H.: Computing power series in polynomial time. Advances in Applied Mathematics 9(1), 40–50 (1988)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Kawamura, A.: Complexity of initial value problems. Fields Institute Communications (to appear)Google Scholar
  11. 11.
    Kawamura, A., Cook, S.: Complexity theory for operators in analysis. ACM Transactions on Computation Theory 4(2), Article 5 (2012)Google Scholar
  12. 12.
    Bournez, O., Graça, D.S., Pouly, A.: Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 170–181. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  13. 13.
    Wagner, K.: The complexity of combinatorial problems with succinct input representation. Acta Informatica 23(3), 325–356 (1986)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Torán, J.: Complexity classes defined by counting quantifiers. Journal of the ACM 38(3), 752–773 (1991)CrossRefGoogle Scholar
  15. 15.
    Kawamura, A.: On function spaces and polynomial-time computability. Dagstuhl Seminar 11411: Computing with Infinite Data (2011)Google Scholar
  16. 16.
    Kawamura, A., Müller, N., Rösnick, C., Ziegler, M.: Uniform polytime computable operators on univariate real analytic functions. In: Proceedings of the Ninth International Conference on Computability and Complexity in Analysis (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Akitoshi Kawamura
    • 1
  • Hiroyuki Ota
    • 1
  • Carsten Rösnick
    • 2
  • Martin Ziegler
    • 2
  1. 1.University of TokyoTokyoJapan
  2. 2.Technische Universität DarmstadtDarmstadtGermany

Personalised recommendations