Abstract
A unified treatment of three point quadrature rules is presented in which the classical rules of mid-point, trapezoidal and Simpson type are recaptured as particular cases. Riemann integrals are approximated for the derivative of the integrand belonging to a variety of norms. The Grüss inequality and a number of variants are also presented which provide a variety of inequalities that are suitatable for numerical implementation. Mappings that are of bounded total variation, Lipschitzian and monotonic are also investigated with relation to Riemann-Stieltjes integrals. Explicit a priori bounds are provided allowing the determination of the partition required to achieve a prescribed error tolerance.
It is demonstrated that with the above classes of functions, the average of a mid-point and trapezoidal type rule produces the best bounds.
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Cerone, P., Dragomir, S.S. (2002). Three Point Quadrature Rules. In: Dragomir, S.S., Rassias, T.M. (eds) Ostrowski Type Inequalities and Applications in Numerical Integration. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2519-4_3
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DOI: https://doi.org/10.1007/978-94-017-2519-4_3
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