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 \({S^2 \subset \mathbb{R}^3}\) 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.
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Dedicated to Götz Alefeld on the occasion of his 70th birthday.
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Chen, X., Frommer, A. & Lang, B. Computational existence proofs for spherical t-designs. Numer. Math. 117, 289–305 (2011). https://doi.org/10.1007/s00211-010-0332-5
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DOI: https://doi.org/10.1007/s00211-010-0332-5