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Numerische Mathematik

, Volume 117, Issue 2, pp 289–305 | Cite as

Computational existence proofs for spherical t-designs

  • Xiaojun Chen
  • Andreas Frommer
  • Bruno Lang
Article

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.

Mathematics Subject Classification (2000)

65H10 65G20 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Applied MathematicsHong Kong Polytechnic UniversityKowloonHong Kong
  2. 2.Department of MathematicsUniversity of WuppertalWuppertalGermany

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