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
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):
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:
Presentation (39 frames) "Towards a General Error Theory of the
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:
Download the full text (16 pages), "Towards a General Error Theory of
the Trapezoidal Rule", to be published by Springer: