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Numerical Integration with Exact Real Arithmetic

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Book cover Automata, Languages and Programming

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1644))

Abstract

We show that the classical techniques in numerical integration (namely the Darboux sums method, the compound trapezoidal and Simpson’s rules and the Gauss-Legendre formulae) can be implemented in an exact real arithmetic framework in which the numerical value of an integral of an elementary function is obtained up to any desired accuracy without any round-off errors. Any exact framework which provides a library of algorithms for computing elementary functions with an arbitrary accuracy is suitable for such an implementation; we have used an exact real arithmetic framework based on linear fractional transformations and have thereby implemented these numerical integration techniques. We also show that Euler’s and Runge-Kutta methods for solving the initial value problem of an ordinary differential equation can be implemented using an exact framework which will guarantee the convergence of the approximation to the actual solution of the differential equation as the step size in the partition of the interval in question tends to zero.

This work is supported by EPSRC.

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Author information

Authors and Affiliations

  1. Department of Computing, Imperial College, 180 Queen’s Gate, London, SW7 2BZ, UK

    Abbas Edalat & Marko Krznarić

Authors
  1. Abbas Edalat
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  2. Marko Krznarić
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Editor information

Editors and Affiliations

  1. Institute for Computer Science, Academy of Sciences of the Czech Republic, Pod vodarenskou vezi 2, 182 07, Prague 8 - Liben, Czech Republic

    Jirí Wiedermann

  2. ILLC-WINS-University of Amsterdam, Plantage Muidergracht 24, 1018, TV, Amsterdam, The Netherlands

    Peter van Emde Boas

  3. BRICS, Department of Computer Science, University of Aarhus, Ny Munkegade, Bldg. 540, DK, 8000, Aarhus C, Denmark

    Mogens Nielsen

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© 1999 Springer-Verlag Berlin Heidelberg

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Edalat, A., Krznarić, M. (1999). Numerical Integration with Exact Real Arithmetic. In: Wiedermann, J., van Emde Boas, P., Nielsen, M. (eds) Automata, Languages and Programming. Lecture Notes in Computer Science, vol 1644. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48523-6_7

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  • DOI: https://doi.org/10.1007/3-540-48523-6_7

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66224-2

  • Online ISBN: 978-3-540-48523-0

  • eBook Packages: Springer Book Archive

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