Computing Integrals of Analytic Functions to High Precision

by Jörg Waldvogel
Seminar for Applied Mathematics
Swiss Federal Institute of Technology ETH
CH-8092 Zürich, Switzerland


We present a set of numerical quadrature algorithms which typically show exponential convergence for analytic integrands, even in the presence of integrable boundary singularities. The algorithms are based on mapping the integration interval onto the entire real axis, together with suitable transformations of the integrand, preferrably to a doubly-exponentially decaying function. The transformed integrals are approximated efficiently by the trapezoidal rule; the approximation error may be analyzed by means of Fourier theory.

This method results in a practicable algorithm for computing analytic integrals to a precision of hundreds - or thousands - of digits. Such high precision may prove meaningful for, e.g., identifying new numbers (defined by integrals) with combinations of known mathematical constants [1, 3]. An elegant, almost fully automated experimental implementation in the language PARI/GP is given.

Download a preliminary version (work in progress, 21 pages): integrals.pdf

Presentation (18 frames) "Computing Integrals of Analytic Functions: A Universal Algorithm with Exponential Convergence".
New Methods for Quadrature. Third Scopes Meeting, ETH Zürich, December 7 - 10, 2006: nis2006.pdf

Presentation (39 frames) "Towards a General Error Theory of the Trapezoidal Rule".
Fourth Scopes Meeting, Hotel Villa List, Sozopol, Bulgaria, September 12 - 18, 2007, and
Approximation and Computation 2008. A meeting dedicated to the 60-th Birthday of Gradimir V. Milovanovic. Nis, Serbia, August 25 - 29, 2008: sozopol.pdf

Download the full text (16 pages), "Towards a General Error Theory of the Trapezoidal Rule", to be published by Springer: nisJoerg.pdf