Computational existence proofs for spherical <Emphasis Type="Italic">t</Emphasis>-designs |
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Authors: | Xiaojun Chen Andreas Frommer Bruno Lang |
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Institution: | 1.Department of Applied Mathematics,Hong Kong Polytechnic University,Kowloon,Hong Kong;2.Department of Mathematics,University of Wuppertal,Wuppertal,Germany |
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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 ì \mathbbR3{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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