Error Bounds for Gaussian Quadrature of Analytic Functions

Abstract

For Gaussian quadrature rules over a finite interval, applied to analytic or meromorphic functions, we develop error bounds from contour integral representations of the remainder term. As in previous work on the subject, we consider both circular and elliptic contours. In contrast with earlier work, however, we attempt to determine exactly where on the contour the kernel of the error functional attains its maximum modulus. We succeed in answering this question for a large class of weight distributions (including all Jacobi weights) when the contour is a circle. In the more difficult case of elliptic contours, we can settle the question for certain special Jacobi weight distributions with parameters $ \pm \frac{1} {2}$, and we provide empirical results for more general Jacobi weights. We further point out that the kernel of the error functional, at any complex point outside the interval of integration, can be evaluated accurately and efficiently by a recursive procedure. The same procedure is useful also to evaluate certain correction terms that arise when poles are present in the integrand. The error bounds obtained are illustrated numerically for two examples—an integral representation for the Bessel function of order zero, and an integral related to the complex exponential integral.