Computation of definite integrals to high precision (typically several hundred digit precision) has emerged as a particularly fruitful tool for experimental mathematics. In many cases, integrals with no known analytic evaluations have been experimentally evaluated (pending subsequent formal proof) by applying techniques such as the PSLQ integer relation algorithm to the output numerical values. In other cases, intriguing linear relations have been found in a class of related integrals, relations which have subsequently been proven as instances of more general results. In this lecture, Bailey will introduce the two principal algorithms used for high-precision integration, namely Gaussian quadrature and tanh-sinh quadrature, with some details on efficient computer implementations. He will also present numerous examples of new mathematical results obtained, in part, by using these methods.

In a subsequent lecture, Bailey will discuss the PSLQ algorithm and give the details of efficient multi-level and parallel implementations.