Computational existence proofs for spherical t-designs

Spherical t-designs provide quadrature rules for the sphere which are exact for polynomials up to degree t. In this paper, we propose a computational algorithm based on interval arithmetic which, for given t, upon successful completion will have proved the existence of a tdesign with (t + 1)2 nodes and will have computed narrow interval enclosures which are known to contain these nodes with mathematical certainty. Since there is no theoretical result which proves the existence of a t-design with (t+ 1)2 nodes for arbitrary t, our method contributes to the theory because it was tested successfully for t = 1, 2, . . . , 100, i.e., for all t considered so far. The t-design is usually not unique; our method aims at finding a well-conditioned one. The method relies on computing an interval enclosure for the zero of a highly nonlinear system of dimension (t+ 1)2. We therefore develop several special approaches which allow us to use interval arithmetic efficiently in this particular situation. The computations were all done using the MATLAB toolbox INTLAB.