Improving the Accuracy of Exponentially Converging Quadratures

Evaluation of one-dimensional integrals arises in many problems in physics and technology. This is most often done using simple quadratures of midpoints, trapezoids and Simpson on a uniform grid. For integrals of periodic functions over the full period, the convergence of these quadratures drastically accelerates and depends on the number of grid steps according to an exponential law. In this paper, new asymptotically accurate estimates of the error of such quadratures are obtained. They take into account the location and multiplicity of the poles of the integrand in the complex plane. A generalization of these estimates is constructed for the case when there is no a priori information about the poles of the integrand. An error extrapolation procedure is described that drastically accelerates the convergence of quadratures.