Towards Using Exact Real Arithmetic for Initial Value Problems

  • Franz Brauße
  • Margarita Korovina
  • Norbert Th. MüllerEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9609)


In the paper we report on recent developments of the iRRAM software [7] for exact real computations. We incorporate novel methods and tools to generate solutions of initial value problems for ODE systems with polynomial right hand sides (PIVP). The algorithm allows the evaluation of the solutions with an arbitrary precision on their complete open intervals of existence. In consequence, the set of operators implemented in the iRRAM software (like function composition, computation of limits, or evaluation of Taylor series) is expanded by PIVP solving.


Computable analysis Taylor models Ordinary differential equations Exact real arithmetic 


  1. 1.
    Kawamura, A., Ota, H., Rösnick, C., Ziegler, M.: Computational complexity of smooth differential equations. Log. Meth. Comput. Sci. 10(1), 1–15 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Makino, K., Berz, M.: Taylor models and other validated functional inclusion methods. Int. J. Pure Appl. Math. 4(4), 379–456 (2003)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Müller, N., Korovina, M.: Making Big Steps in Trajectories. In: EPTCS (2010)Google Scholar
  4. 4.
    Müller, N.T.: Polynomial time computation of Taylor series. In: Proceedings of the 22th JAIIO - Panel 1993, Part 2, pp. 259–281, Buenos Aires (1993)Google Scholar
  5. 5.
    Müller, N.T.: Constructive aspects of analytic functions. In: Ko, K.I., Weihrauch, K. (eds.) Computability and Complexity in Analysis. Informatik Berichte, vol. 190, pp. 105–114. FernUniversität Hagen (September 1995), CCA Workshop, Hagen, 19–20 August 1995Google Scholar
  6. 6.
    Müller, N.T.: Towards a real real RAM: a prototype using C++. In: Ko, K.I., Mller, N., Weihrauch, K. (eds.) Computability and Complexity in Analysis, pp. 59–66. Universität Trier, Second CCA Workshop, Trier, 22–23 August 1996Google Scholar
  7. 7.
    Müller, N.T.: The iRRAM: exact arithmetic in C++. In: Blank, J., Brattka, V., Hertling, P. (eds.) CCA 2000. LNCS, vol. 2064, p. 222. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Nedialkov, N.S.: VNODE-LP — a validated solver for initial value problems inordinary differential equations. Technical report, CAS-06-06-NN, Department of Computingand Software, McMaster University, Hamilton, Ontario, L8S 4K1 (2006)Google Scholar
  9. 9.
    Nedialkov, N.S., Jackson, K.R., Corliss, G.F.: Validated solutions of initial value problems for ordinary differential equations. Appl. Math. Comput. 105(1), 21–68 (1999)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Parker, G.E., Sochacki, J.S.: Implementing the Picard iteration. neural, parallel. Sci. Comput. 4(1), 97–112 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Pouly, A., Graça, D.S.: Computational complexity of solving elementary differential equations over unbounded domains. CoRR abs/1409.0451 (2014)Google Scholar
  12. 12.
    Teschl, G.: Ordinary Differential Equations and Dynamical Systems. Graduate Studies in Mathematics. American Mathematical Society, Providence (2012)CrossRefzbMATHGoogle Scholar
  13. 13.
    Weihrauch, K.: Computable Analysis: An Introduction. Springer-Verlag New York, Inc., Secaucus (2000)CrossRefzbMATHGoogle Scholar
  14. 14.
    Zimmermann, P.: Reliable computing with GNU MPFR. In: Fukuda, K., Hoeven, J., Joswig, M., Takayama, N. (eds.) ICMS 2010. LNCS, vol. 6327, pp. 42–45. Springer, Heidelberg (2010)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Franz Brauße
    • 1
  • Margarita Korovina
    • 2
  • Norbert Th. Müller
    • 1
    Email author
  1. 1.Abteilung InformatikwissenschaftenUniversität TrierTrierGermany
  2. 2.A.P. Ershov Institute of Informatics SystemsNovosibirskRussia

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