On the computation of nonnegative quadrature weights on the sphere

We compute quadrature weights for scattered nodes on the two-dimensional unit-sphere, which are exact for spherical polynomials of high degree N. Different algorithms are proposed and numerical examples show that we can compute nonnegative quadrature weights if approximately 4N2/3 well distributed nodes are used. We compare these results with theoretical statements which guarantee nonnegative quadrature weights. The proposed algorithms are based on fast spherical Fourier algorithms for arbitrary nodes which are publicly available. Numerical experiments are presented to demonstrate that we are able to compute quadrature weights for circa 1.5 million nodes which are exact for spherical polynomials up to N=1024.