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 .


Computational Complexity Real Function Limited Feedback String Function Uniform Family 
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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

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