On the efficient computation of high-dimensional integrals and the approximation by exponential sums

The approximation of the functions 1/x and \(1/\sqrt{x}\) by exponential sums enables us to evaluate some high-dimensional integrals by products of one-dimensional integrals. The degree of approximation can be estimated via the study of rational approximation of the square root function. The latter has interesting connections with the Babylonian method and Gauss’ arithmetic-geometric process.