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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Automatic differentiation of numerical integration algorithms
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by Peter Eberhard and Christian Bischof PDF
Math. Comp. 68 (1999), 717-731 Request permission


Automatic differentiation (AD) is a technique for automatically augmenting computer programs with statements for the computation of derivatives. This article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations (ODEs), in particular, the ramifications of the fact that AD is applied not only to the solution of such an algorithm, but to the solution procedure itself. This subtle issue can lead to surprising results when AD tools are applied to variable-stepsize, variable-order ODE integrators. The computation of the final time step plays a special role in determining the computed derivatives. We investigate these issues using various integrators and suggest constructive approaches for obtaining the desired derivatives.
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Additional Information
  • Peter Eberhard
  • Affiliation: Institute B of Mechanics, University of Stuttgart, 70550 Stuttgart, Germany
  • Email:
  • Christian Bischof
  • Affiliation: Rheinisch-Westfälische Technische Hochschule Aachen, Seffenter Weg 23, D-52056 Aachen, Germany
  • Email:
  • Received by editor(s): November 11, 1996
  • Received by editor(s) in revised form: July 24, 1997
  • Additional Notes: This work was partially completed while the first author was visiting the Department of Mechanical Engineering at the University of California at Berkeley supported by the German Research Council (DFG) under grant EB195/1-1.
    The second author was supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under contract W-31-109-Eng-38.
  • © Copyright 1999 American Mathematical Society
  • Journal: Math. Comp. 68 (1999), 717-731
  • MSC (1991): Primary 34A12, 65L05, 65L06; Secondary 68N99
  • DOI:
  • MathSciNet review: 1613703