Error-bounds for the evaluation of integrals by the Euler-Maclaurin formula and by Gauss-type formulae

Abstract

where f(x) is even, can be evaluated with surprising accuracy by means of the trapezoidal-sum formula. It is natural to anticipate that a more general result can be obtained when the integrand is not restricted to the form given above; the generalization was easily obtained by contour integration. The guiding idea which we employ in obtaining error-bounds is extremely simple: we express the error-term of an approximate integration by a contour integral and then choose a contour (among those which enclose the singularities of the integrand) on which the error-term is simply and easily bounded. Most often, it turns out that minimizing the absolute value of the error-term integrand is quite effective; and, in general, we have preferred methods for obtaining error-bounds which can be used and extended without undue expenditure of time by the staff of a computing laboratory. Subsequently, it was found that the technique of minimizing the integrand had been used earlier by Davis [2] in a paper whose contents somewhat overlap the contents of the present paper. Further trial showed that the guiding idea could be used to obtain useful error-bounds for Gauss-type formulae. These bounds are reported in Section 3. In the main, the paper is concerned with the Euler-Maclaurin formula and extensions