TW 308

R. Cools and E. Novak
Spherical product algorithms and the integration of smooth functions with one singular point

Abstract

We consider the problem of numerical integration for multivariate functions with respect to a radial symmetric weight. We prove that suitable spherical product algorithms have the optimal rate of convergence n^-k/d for C^k-functions. We also study classes of integrands with a singularity that are C^k outside the origin. Standard algorithms have high cost for such functions, because they require that the function is smooth everywhere. We construct suitably modified spherical product algorithms with optimal rate of convergence n^-k/d also in this case. In the compact case we can use modified spherical product Gauss formulas with a nonalgebraic degree of precision.