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Two adaptive Gauss-Legendre type algorithms for the verified computation of definite integrals

Два адаптивных алгоритма типа Гаусса-Лежандра для верифицируемого вычисления определенных интегралов

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Reliable Computing

Abstract

We propose a two algorithms for computation of (sharp) enclosures of definite interevals: alocal adaptive algorithm (LAA) and aglobal adaptive algorithm (GAA). Both algorithms are based on Gauss-Legendre quadrature. Error terms are bounded using automatic differentiation in combination with interval evaluations.

Several numerical examples are presented; these examples include comparison with an adaptive interval Romberg scheme.

Abstract

Предлагаются два алгоритма для вычисления (тесных) включений определенных интегралов: локальный адаптивный алгоритм и глобальный адаптивный алгоритм. Оба алгоритма основаны на квадратуре Гаусса-Лежандра. Члены, характеризующие погрещность, находятся с помощью автоматического дифференцирования в сочетании с интервальными оценками.

Представлено несколько численных примеров, которые включают сравнение с адаптивной интервальной схемой Ромберга.

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Authors and Affiliations

  1. Institut für Wissenschaftliches Rechnen und Mathematische Modellbildung (IWRMM), Germany

    W. Krämer

  2. Institut für Angewandte Mathematik, Universität Karlsruhe, 76128, Karlsruhe, Germany

    S. Wedner

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  1. W. Krämer
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  2. S. Wedner
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© W. Krämer, S. Wedner, 1996

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Krämer, W., Wedner, S. Two adaptive Gauss-Legendre type algorithms for the verified computation of definite integrals. Reliable Comput 2, 241–253 (1996). https://doi.org/10.1007/BF02391698

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  • DOI: https://doi.org/10.1007/BF02391698

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