Hybrid symbolic-numeric multiple integration and other applications of tensor product series
Keith Geddes, Symbolic Computation Group, University of Waterloo.
Abstract: A hybrid symbolic-numeric method for the fast and accurate evaluation of definite integrals in multiple dimensions has been developed based on approximating the multivariate integrand by tensor product series. This method is well-suited for two classes of problems: (1) analytic integrands over general regions in two dimensions, and (2) families of analytic integrands with special algebraic structure over hyperrectangular regions in higher dimensions. 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. Our approach overcomes this difficulty by using a tensor product series with a modest number of terms to construct an accurate approximation of the integrand. The partial separation of variables achieved in this way reduces the original integral to a manageable bilinear combination of integrals of half the original dimension. We continue halving the dimensions recursively until obtaining one-dimensional integrals, which are then computed by standard numeric or symbolic techniques. We have also applied tensor product series expansions as a technique for deriving and proving identities for bivariate functions, as presented at the Calculemus 2006 conference. Whereas our method for multidimensional integration is based on approximating the integrand by a tensor product series, the Calculemus paper exploits the concept of expanding a bivariate function in a tensor product series, specifically focusing on cases where the series terminates yielding a tensor product identity. The result is an algorithm for the automatic derivation and proof of identities such as the formula for cos(x+y) and similar identities for various elementary and special functions. This is Joint work with Frederick Chapman.