Numerische Mathematik

, Volume 117, Issue 2, pp 289–305

# 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.

65H10 65G20

## References

1. 1.
Alefeld G., Herzberger J.: Introduction to Interval Computations. Computer Science and Applied Mathematics. Academic Press, New York (1983)Google Scholar
2. 2.
Bannai E., Bannai E.: A survey on spherical designs and algebraic combinatorics on spheres. Eur. J. Comb. 30, 1392–1425 (2009)
3. 3.
Böhm H.: Evaluation of arithmetic expressions with maximum accuracy. In: Kulisch, U., Miranker, W.L. (eds) A New Approach to Scientific Computing, pp. 121–137. Academic Press, New York (1983)Google Scholar
4. 4.
Chen X., Womersley R.S.: Existence of solutions to systems of underdetermined equations and spherical designs. SIAM J. Numer. Anal. 44, 2326–2341 (2006)
5. 5.
Delsarte P., Goethals J.M., Seidel J.J.: Spherical codes and designs. Geom. Dedicata 6, 363–388 (1977)
6. 6.
Frommer, A., Lang, B.: Fast and accurate multi-argument interval evaluation of polynomials. In: Proceedings of the 12th GAMM—IMACS international symposium on scientific computing, computer arithmetic and validated numerics (SCAN 2006), p. 31, Los Alamitos, CA, USA, 2006. IEEE Computer SocietyGoogle Scholar
7. 7.
Golub G.H., Van Loan C.F.: Matrix Computations. 3rd edn. The Johns Hopkins Univ. Press, Baltimore (1996)
8. 8.
Hardin R.H., Sloane N.J.A.: McLaren’s improved snub cube and other new spherical designs in three dimensions. Discrete Comput. Geom. 15, 429–441 (1996)
9. 9.
Kearfott, R.B., Nakao, M.T., Neumaier, A., Rump, S.M., Shary, S.P., van Hentenryck, P.: Standardized notation in interval analysis (2005). http://www.mat.univie.ac.at/~neum/ms/notation.pdf
10. 10.
Klatte R., Kulisch U.W., Wiethoff A., Lawo Ch., Rauch M.: C-XSC. A C++ Class Library for Extended Scientific Computing. Springer, Berlin (1993)
11. 11.
Korevaar J., Meyers J.L.H.: Spherical Faraday cage for the case of equal point charges and Chebyshev-type quadrature on the sphere. Integral Transform. Spec. Funct. 1, 105–117 (1993)
12. 12.
Krämer, W., Hofschuster, W.: C-XSC 2.0: A C++ library for extended scientific computing. In: Alt, R., Frommer, A., Kearfott, R.B., Luther, W. (eds.) Numerical Software With Result Verification. International Dagstuhl Seminar, Dagstuhl Castle, Germany, January 19–24, 2003. Revised papers. Lecture Notes in Computer Science, vol. 2991, pp. 15–35. Springer, Berlin (2004)Google Scholar
13. 13.
Krämer W., Popova E.D.: Zur Berechnung von verlässlichen Außen- und Inneneinschließungen bei parameterabhängigen linearen Gleichungssystemen. Proc. Appl. Math. Mech. 4, 670–671 (2004)
14. 14.
Krawczyk R.: Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing 4, 187–201 (1969)
15. 15.
Popova E.D.: Parametric interval linear solver. Numer. Algorithms 37, 345–356 (2004)
16. 16.
Rump S.M.: Solving algebraic problems with high accuracy. In: Kulisch, U., Miranker, W.L. (eds) A New Approach to Scientific Computation, pp. 51–120. Academic Press, New York (1983)Google Scholar
17. 17.
Rump, S.M.: Verification methods for dense and sparse systems of equations. In: Herzberger, J. (ed.) Topics in Validated Computations, Stud. Comput. Math., vol. 5, pp. 63–135. Elsevier, Amsterdam (1994)Google Scholar
18. 18.
Rump S.M.: Expansion and estimation of the range of nonlinear functions. Math. Comput. 65(216), 1503–1512 (1996)
19. 19.
Rump S.M.: INTLAB—interval laboratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluwer, Dordrecht (1999)Google Scholar
20. 20.
Seymour P.D., Zaslavsky T.: Averaging sets: a generalization of mean values and spherical designs. Adv. Math. 52, 213–240 (1984)
21. 21.
Sloan I.H., Womersley R.S.: Extremal systems of points and numerical integration on the sphere. Adv. Comp. Math. 21, 102–125 (2004)
22. 22.
Sloan I.H., Womersley R.S.: A variational characterization of spherical designs. J. Approx. Theory 159, 308–318 (2009)
23. 23.
Womersley, R.S.: Interpolation and cubature on the sphere. http://web.maths.unsw.edu.au/~rsw/Sphere/Extremal/New/index.html