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International Conference on Mathematical Aspects of Computer and Information Sciences

MACIS 2015: Mathematical Aspects of Computer and Information Sciences pp 469–473Cite as

Rigorous Numerical Computation of Polynomial Differential Equations Over Unbounded Domains

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9582)

Abstract

In this abstract we present a rigorous numerical algorithm which solves initial-value problems (IVPs) defined with polynomial differential equations (i.e. IVPs of the type \(y'=p(t,y)\), \(y(t_0)=y_0\), where p is a vector of polynomials) for any value of t. The inputs of the algorithm are the data defining the initial-value problem, the time T at which we want to compute the solution of the IVP, and the maximum allowable error \(\varepsilon >0\). Using these inputs, the algorithm will output a value \(\tilde{y}_T\) such that \(\Vert \tilde{y}_T-y(T)\Vert \le \varepsilon \) in time polynomial in T, \(-\log \varepsilon \), and in several quantities related to the polynomial IVP.

Keywords

  • Polynomial Ordinary Differential Equations
  • Rigorous Numerical Calculations
  • Unbounded Domains
  • Maximum Allowable Error
  • Theoretical Computer Science Literature

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Atkinson, K.E.: An Introduction to Numerical Analysis, 2nd edn. Wiley, New York (1989)

    MATH  Google Scholar 

  2. Bostan, A., Chyzak, F., Ollivier, F., Salvy, B., Schost, É., Sedoglavic, A.: Fast computation of power series solutions of systems of differential equations. In: SODA 2007, pp. 1012–1021, January 2007

    Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Kawamura, A.: Lipschitz continuous ordinary differential equations are polynomial-space complete. Comput. Complex. 19(2), 305–332 (2010)

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MATH  Google Scholar 

  6. Lorenz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    CrossRef  Google Scholar 

  7. Müller, N., Moiske, B.: Solving initial value problems in polynomial time. In: Proceedings of 22 JAIIO - PANEL 1993, Part 2, pp. 283–293 (1993)

    Google Scholar 

  8. Pouly, A.: Continuous Models of Computation: From Computability to Complexity. Ph.D. thesis, Ecole Polytechnique/Universidade do Algarve (2015)

    Google Scholar 

  9. Warne, P.G., Warne, D.P., Sochacki, J.S., Parker, G.E., Carothers, D.C.: Explicit a-priori error bounds and adaptive error control for approximation of nonlinear initial value differential systems. Comput. Math. Appl. 52(12), 1695–1710 (2006). http://dx.doi.org/10.1016/j.camwa.2005.12.004

    CrossRef  MathSciNet  MATH  Google Scholar 

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Acknowledgments

D. Graça was partially supported by Fundação para a Ciência e a Tecnologia and EU FEDER POCTI/POCI via SQIG - Instituto de Telecomunicações through the FCT project UID/EEA/50008/2013.

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Authors and Affiliations

  1. LIX, Ecole Polytechnique, 91128, Palaiseau Cedex, France

    Olivier Bournez & Amaury Pouly

  2. CEDMES/FCT, Universidade do Algarve, C. Gambelas, 8005-139, Faro, Portugal

    Daniel S. Graça & Amaury Pouly

  3. SQIG/Instituto de Telecomunicações, Lisbon, Portugal

    Daniel S. Graça

Authors
  1. Olivier Bournez
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  2. Daniel S. Graça
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  3. Amaury Pouly
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Corresponding author

Correspondence to Daniel S. Graça .

Editor information

Editors and Affiliations

  1. Wilfrid Laurier University, Waterloo, Ontario, Canada

    Ilias S. Kotsireas

  2. Hamburg University of Technology, Hamburg, Germany

    Siegfried M. Rump

  3. New York University, New York, New York, USA

    Chee K. Yap

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Bournez, O., Graça, D.S., Pouly, A. (2016). Rigorous Numerical Computation of Polynomial Differential Equations Over Unbounded Domains. In: Kotsireas, I., Rump, S., Yap, C. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2015. Lecture Notes in Computer Science(), vol 9582. Springer, Cham. https://doi.org/10.1007/978-3-319-32859-1_40

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  • DOI: https://doi.org/10.1007/978-3-319-32859-1_40

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-32858-4

  • Online ISBN: 978-3-319-32859-1

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