Computing Narrow Inclusions for Definite Integrals.
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WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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The ACRITH computer program is used to implement algorithms which give narrow inclusions for definite integrals. Inclusions are obtained from familiar numerical quadrature formulas such as Gaussian or Newton-Cotes with remainder terms, or from the term-by-term integration of Taylor polynomials with remainder terms. Inclusions for the remainder terms are computed using automatic differentiation. The inclusions for the remainder terms are computed using automatic differentiation. The inclusions are valid if the integrand or the endpoints of the interval of integration are real-or interval-valued. Interval inclusions which contain only a few machine numbers are achieved by using ACRITHs accurate scalar product, by using order and subinterval adaptation, and by using special devices such as intersection of several estimates. Numerical examples show that such narrow, guaranteed bounds require about four times as long to compute as the estimates computed by the routine QAGS from QUADPACK. Author
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