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Trapezoidal-type rules from an inequalities point of view. (English) Zbl 0966.26014
Anastassiou, George (ed.), Handbook of analytic-computational methods in applied mathematics. Boca Raton, FL: Chapman & Hall/CRC. 65-134 (2000).
Summary: This chapter investigates trapezoidal-type rules and obtains explicit bounds through the use of a Peano kernel approach and the modern theory of inequalities. Both Riemann-Stieltjes and Riemann integrals are evaluated with a variety of assumptions about the integrand enabling the characterization of the bound in terms of a variety of norms. Perturbed quadrature rules are obtained through the use of Grüss, Chebychev and Lupaş inequalities, producing a variety of tighter bounds. The implementation is demonstrated through the investigation of a variety of composite rules based on inequalities developed. The analysis allows the determination of the partition required that would assure that the accuracy of the result would be within a prescribed error tolerance.
MSC:
26D15Inequalities for sums, series and integrals of real functions
65D32Quadrature and cubature formulas (numerical methods)
65D30Numerical integration
41A55Approximate quadratures
26A46Absolutely continuous functions (one real variable)