Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/20621
Title: Computational existence proofs for spherical t-designs
Authors: Chen, X 
Frommer, A
Lang, B
Issue Date: 2011
Source: Numerische mathematik, 2011, v. 117, no. 2, p. 289-305 How to cite?
Journal: Numerische Mathematik 
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 S2 ⊂ ℝ3 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.
URI: http://hdl.handle.net/10397/20621
ISSN: 0029-599X
DOI: 10.1007/s00211-010-0332-5
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