Skip to main content

Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains

  • Conference paper
Mathematical Foundations of Computer Science 2011 (MFCS 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6907))

Abstract

In this paper we consider the computational complexity of solving initial-value problems defined with analytic ordinary differential equations (ODEs) over unbounded domains of ℝn and ℂn, 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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Ko, K.I.: Computational Complexity of Real Functions. Birkhäuser, Basel (1991)

    Google Scholar 

  2. Shannon, C.E.: Mathematical theory of the differential analyzer. J. Math. Phys. 20, 337–354 (1941)

    MathSciNet  MATH  Google Scholar 

  3. Bush, V.: The differential analyzer. A new machine for solving differential equations. J. Franklin Inst. 212, 447–488 (1931)

    Google Scholar 

  4. Graça, D.S., Costa, J.F.: Analog computers and recursive functions over the reals. J. Complexity 19(5), 644–664 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Demailly, J.-P.: Analyse Numérique et Equations Différentielles. Presses Universitaires de Grenoble (1991)

    Google Scholar 

  9. Smith, W.D.: Church’s thesis meets the N-body problem. Applied Mathematics and Computation 178(1), 154–183 (2006)

    Article  MathSciNet  Google Scholar 

  10. Perko, L.: Differential Equations and Dynamical Systems, 3rd edn. Springer, Heidelberg (2001)

    MATH  Google Scholar 

  11. Ruohonen, K.: An effective Cauchy-Peano existence theorem for unique solutions. Internat. J. Found. Comput. Sci. 7(2), 151–160 (1996)

    Article  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. Müller, N.T., Korovina, M.V.: Making big steps in trajectories. Electr. Proc. Theoret. Comput. Sci. 24, 106–119 (2010)

    Article  Google Scholar 

  15. Birkhoff, G., Rota, G.C.: Ordinary Differential Equations, 4th edn. John Wiley, Chichester (1989)

    Google Scholar 

  16. Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)

    MATH  Google Scholar 

  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. Grzegorczyk, A.: On the definitions of computable real continuous functions. Fund. Math. 44, 61–71 (1957)

    MathSciNet  MATH  Google Scholar 

  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. Weihrauch, K.: Computable Analysis: an Introduction. Springer, Heidelberg (2000)

    MATH  Google Scholar 

  21. Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Springer, Heidelberg (1989)

    MATH  Google Scholar 

  22. Ko, K.I., Friedman, H.: Computational complexity of real functions. Theoret. Comput. Sci. 20, 323–352 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  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. Graça, D.S., Campagnolo, M.L., Buescu, J.: Computability with polynomial differential equations. Adv. Appl. Math. 40(3), 330–349 (2008)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag GmbH Berlin Heidelberg

About this paper

Cite this paper

Bournez, O., Graça, D.S., Pouly, A. (2011). Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains. In: Murlak, F., Sankowski, P. (eds) Mathematical Foundations of Computer Science 2011. MFCS 2011. Lecture Notes in Computer Science, vol 6907. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22993-0_18

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-22993-0_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22992-3

  • Online ISBN: 978-3-642-22993-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics