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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Rigorous uniform approximation of D-finite functions using Chebyshev expansions
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by Alexandre Benoit, Mioara Joldeş and Marc Mezzarobba PDF
Math. Comp. 86 (2017), 1303-1341

Abstract:

A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of such methods in the context of rigorous computing (where we need guarantees on the accuracy of the result), and from the complexity point of view.

It is well known that the order-$n$ truncation of the Chebyshev expansion of a function over a given interval is a near-best uniform polynomial approximation of the function on that interval. In the case of solutions of linear differential equations with polynomial coefficients, the coefficients of the expansions obey linear recurrence relations with polynomial coefficients. Unfortunately, these recurrences do not lend themselves to a direct recursive computation of the coefficients, owing among other things to a lack of initial conditions.

We show how they can nevertheless be used, as part of a validated process, to compute good uniform approximations of D-finite functions together with rigorous error bounds, and we study the complexity of the resulting algorithms. Our approach is based on a new view of a classical numerical method going back to Clenshaw, combined with a functional enclosure method.

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Additional Information
  • Alexandre Benoit
  • Affiliation: Éducation nationale, France
  • Email: alexandrebenoit@yahoo.fr
  • Mioara Joldeş
  • Affiliation: CNRS, LAAS, 7 Avenue du Colonel Roche, 31077 Toulouse, Cedex 4, France
  • MR Author ID: 931466
  • Email: joldes@laas.fr
  • Marc Mezzarobba
  • Affiliation: CNRS, UMR 7606, LIP6, F-75005, Paris, France – and – Sorbonne Universités, UPMC Univ Paris 06, UMR 7606, LIP6, F-75005, Paris, France
  • MR Author ID: 904789
  • Email: marc@mezzarobba.net
  • Received by editor(s): July 10, 2014
  • Received by editor(s) in revised form: November 7, 2015, and November 22, 2015
  • Published electronically: October 20, 2016
  • Additional Notes: This research was partly supported by the Austria Science Fund (FWF) grants P22748-N18 and Y464-N18, and previously by the MSR-Inria joint research center
    This work is in the public domain. As such, it is not subject to copyright. In jurisdictions where it is not possible to consider this work as released into the public domain, the authors grant any entity the right to use this work for any purpose, without any conditions, unless such conditions are required by law.
  • © Copyright 2016 by the authors
  • Journal: Math. Comp. 86 (2017), 1303-1341
  • MSC (2010): Primary 68W30, 33C45, 65L05, 65G20
  • DOI: https://doi.org/10.1090/mcom/3135
  • MathSciNet review: 3614019