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International Conference on Interactive Theorem Proving

ITP 2012: Interactive Theorem Proving pp 377–392Cite as

Numerical Analysis of Ordinary Differential Equations in Isabelle/HOL

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7406))

Abstract

Many ordinary differential equations (ODEs) do not have a closed solution, therefore approximating them is an important problem in numerical analysis. This work formalizes a method to approximate solutions of ODEs in Isabelle/HOL.

We formalize initial value problems (IVPs) of ODEs and prove the existence of a unique solution, i.e. the Picard-Lindelöf theorem. We introduce generic one-step methods for numerical approximation of the solution and provide an analysis regarding the local and global error of one-step methods.

We give an executable specification of the Euler method as an instance of one-step methods. With user-supplied proofs for bounds of the differential equation we can prove an explicit bound for the global error. We use arbitrary-precision floating-point numbers and also handle rounding errors when we truncate the numbers for efficiency reasons.

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Author information

Authors and Affiliations

  1. Institut für Informatik, Technische Universität München, Germany

    Fabian Immler & Johannes Hölzl

Authors
  1. Fabian Immler
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  2. Johannes Hölzl
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Princeton University, 35 Olden Street, 08540, Princeton, NJ, USA

    Lennart Beringer

  2. School of Electrical Engineering and Computer Science, University of Ottawa, 800 King Edward Ave., K1N 6N5, Ottawa, ON, Canada

    Amy Felty

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Immler, F., Hölzl, J. (2012). Numerical Analysis of Ordinary Differential Equations in Isabelle/HOL. In: Beringer, L., Felty, A. (eds) Interactive Theorem Proving. ITP 2012. Lecture Notes in Computer Science, vol 7406. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32347-8_26

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  • DOI: https://doi.org/10.1007/978-3-642-32347-8_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-32346-1

  • Online ISBN: 978-3-642-32347-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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