Computational existence proofs for spherical t-designs

Abstract

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 t-design with (t + 1) 2 nodes on the unit sphere S 2 ⊂ R 3 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. 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. Mathematics Subject Classification (2000) 65H10 · 65G20Dedicated to Götz Alefeld on the occasion of his 70th birthday.

Well Conditioned Spherical Designs for Integration and Interpolation on the Two-Sphere

“…Chen and Womersley [7] and then Chen, Frommer, and Lang [6] verified that a spherical t-design exists in a neighborhood of an extremal system. This leads to the idea of extremal spherical t-designs, which first appeared in [7] for the special case N = (t + 1)…”

“…Interval methods [1,6,25] are then used to prove the existence of a well conditioned true spherical t-design in a narrow interval and to place relatively close upper and lower bounds on the determinant of the matrix H t (X N ) over the interval.…”

“…The condition in Proposition 2.5 that G t (X N ) is nonsingular is essential, as shown by Example 2 of [6]. In that example, t = 1, and X 4 consists of the following four points:…”

“…Since then, spherical t-designs have been studied extensively [2,3,6,7,13,17,29,30,32]. In 2009, Bannai and Bannai gave a comprehensive survey of research on spherical t-designs in the last three decades [4].…”

“…However, from the work of [6] we know that, for d = 2, spherical t-designs with (t + 1) 2 points do exist for all degrees t up to 100, so extremal spherical t-designs are well defined up to at least t = 100, but until now no attempt has been made to compute them. This range of t values is large enough to persuade us of the usefulness of the definition.…”