We present a new hybrid symbolic-numeric method for the accurate evaluation of definite integrals in multiple dimensions. The theory of multivariate approximation via natural tensor product series has been developed in the doctoral thesis by Chapman, who named such series expansions Geddes series in honour of his thesis supervisor. In this paper, we give algorithms for the generation of Geddes series expansions and explore their application to multiple integration. Our new adaptive integration algorithm is effective both in high dimensions and with high accuracy.
Current numerical multiple integration methods either become very slow or yield only low accuracy in high dimensions, due to the necessity to sample the integrand at a very large number of points. The key idea in our approach is to exploit the fact that truncating a Geddes series expansion of the integrand accurately approximates the integrand by a natural tensor product, such that a separation of variables is achieved. Thus we are able to reduce the original integral to a combination of integrals whose dimension is half of the original. At the base, one-dimensional integrals must be computed by either numeric or symbolic techniques.