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On optimal adaptive quadratures for automatic integration

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Abstract

In this paper, the problem of automatic integration is investigated. Quadratures that compute the integral of an r times differentiable function with the assumption that the rth derivative is positive within precision \(\varepsilon >0\) are constructed. A rigorous analysis of these quadratures is presented. It turns out that the mesh selection procedure proposed in this paper is optimal i.e., it uses the minimal number of function evaluations. It is also shown how to adapt the quadratures to the case where the assumption that the rth derivative has a constant sign is not fulfilled. The theoretical results are illustrated and confirmed with numerical tests.

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Acknowledgements

We would like to thank the referees and the editors for a thorough reading and for many useful comments and suggestions that helped us to improve this paper.

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Authors and Affiliations

  1. Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, paw. A3/A4, III p., pok. 301, 30-059, Kraków, Poland

    Maciej Goćwin

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  1. Maciej Goćwin
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Correspondence to Maciej Goćwin.

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Communicated by Michiel E. Hochstenbach.

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This research was partly supported by the Polish Ministry of Science and Higher Education and by the National Science Centre, Poland, under Project 2017/25/B/ST1/00945.

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Goćwin, M. On optimal adaptive quadratures for automatic integration. Bit Numer Math 61, 411–439 (2021). https://doi.org/10.1007/s10543-020-00831-2

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  • DOI: https://doi.org/10.1007/s10543-020-00831-2

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