Asymmetric cubature formulae with few points in high dimension for symmetric measures

SIAM Journal on Numerical Analysis

Volumn 42, Issue 1, 2004, Pages 209-227

  Victoir, Nicolas ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Cubature formulae; Orthogonal array; T design

Indexed keywords

APPROXIMATION THEORY; COMBINATORIAL MATHEMATICS; FUNCTIONS; INTEGER PROGRAMMING; INTEGRATION; NUMBER THEORY; POLYNOMIALS; PROBLEM SOLVING;

CUBATURE FORMULAE; NATURAL NUMBER; ORTHOGONAL ARRAY; T-DESIGN;

NUMERICAL METHODS;

EID: 1542771653     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142902407952     Document Type: Article

References (30)

1. H. BERENS, H. J. SCHMID, AND Y. XU, Multivariate Gauasian cubature formulae, Arch. Math. (Basel), 64 (1995), pp. 26-32.
2. T. BETH, D. JUNGNICKEL, AND H. LENZ, Design Theory, Vol. I, 2nd ed., Encyclopedia Math. Appl. 78, Cambridge University Press, Cambridge, UK, 1999.
3. T. BETH, D. JUNGNICKEL, AND H. LENZ, Design Theory, Vol. II, 2nd ed., Encyclopedia Math. Appl. 78, Cambridge University Press, Cambridge, UK, 1999.
4. J. BIERBRAUER, S. BLACK, AND Y. EDEL, Some t-homogeneous sets of permutations. Des. Codes Cryptogr., 9 (1996), pp. 29-38.
5. R. COOLS, A survey of methods for constructing cubature formulae, in Numerical Integration (Bergen, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 357, T. O. Espelid and A. Genz, eds., Kluwer, Dordrecht, 1992, pp. 1-24.
6. R. COOLS, Constructing cubature formulae: The science behind the art, in Acta Numerica 6, A. Iserles, ed., Cambridge University Press, Cambridge, UK, 1997, pp. 1-54.
7. R. COOLS, Monomial cubature rules since "Stroud": A compilation - part 2, J. Comput. Appl. Math., 112 (1999), pp. 21-27.
8. R. COOLS AND P. RABINOWITZ, Monomial cubature rules since "Stroud": A compilation, J. Comput. Appl. Math., 48 (1993), pp. 309-336.
9. P. J. DAVIS AND P. RABINOWITZ, Methods of Numerical Integration, 2nd ed., Comput. Sci. Appl. Math., Academic Press, Orlando, 1984.
10. P. DELSARTE, An Algebraic Approach to the Association Schemes of Coding Theory, Philips Res. Rep. Suppl. 10, 1973.
11. L. N. DOBRODEEV, Cubature rules with equal coefficients for integrating functions with respect to symmetric domains, Zh. Vychisl. Mat. Mat. Fiz, 18 (1978), pp. 846-852 (in Russian); USSR Comput. Math. Math. Phys., 18(4) (1979), pp. 27-34 (in English).
12. L. N. DOBRODEEV, Cubature rules with equal coefficients for integrating functions with respect to symmetric domains, Zh. Vychisl. Mat. Mat. Fiz, 18 (1978), pp. 846-852 (in Russian); USSR Comput. Math. Math. Phys., 18(4) (1979), pp. 27-34 (in English).
13. M. HALL, JR., Combinatorial Theory, 2nd ed., Wiley-Intersci. Ser. Discrete Math., John Wiley, New York, 1986.
14. 4-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), pp. 301-319.
15. A. S. HEDAYAT, N. J. A. SLOANE, AND J. STUFKEN, Orthogonal Arrays. Theory and Applications, Springer Ser. Statist., Springer-Verlag, New York, 1999.
16. Y. J. IONIN, Applying balanced generalized weighing matrices to construct block designs, Electron. J. Combin., 8 (2001), no. 1, Research Paper 12, 15 pp.
17. Z. JANKI AND T. VAN TRUNG, The existence of a symmetric block design for (70, 24, 8), Mitt. Math. Sem. Giessen, 165 (1984), pp. 17-18.
18. W. M. KANTOR, k-homogeneous groups, Math. Z., 124 (1972), pp. 261-265.
19. A. R. KROMMER AND C. W. UEBERHUBER, Computational Integration, SIAM, Philadelphia, 1998.
20. I. G. MACDONALD, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford University Press, Oxford, UK, 1995.
21. F. J. MACWILLIAMS AND N. J. A. SLOANE, The Theory of Error-Correcting Codes. I, North-Holland Math. Library 16, North-Holland, Amsterdam, 1977.
22. C. J. MITCHELL, An infinite family of symmetric designs, Discrete Math., 26 (1979), pp. 247-250.
23. H. M. MÖLLER, Lower bounds for the number of nodes in cubature formulae, in Numerische Integration (Tagung, Math. Forschungsinst., Oberwolfach, 1978), Internat. Ser. Numer. Math. 45, Birkhäuser, Basel, 1979, pp. 221-230.
24. I. P. MYSOVSKIKH, Interpolatory Cubature Formulas, Nauka, Moscow, Leningrad, 1981 (in Russian).
25. K. NOMURA, On t-homogeneous permutation sets, Arch. Math. (Basel), 44 (1985), pp. 485-487.
26. M. PUTINAR, A note on Tchakaloff's theorem, Proc. Amer. Math. Soc., 125 (1997), pp. 2409-2414.
27. S. L. SOBOLEV, Cubature formulas on the sphere invariant under finite groups of rotation, Soviet Math. Dokl., 3 (1962), pp. 1307-1310.
28. E. SPENCE, A new family of symmetric 2-(v, k, λ) block designs, European J. Combin., 14 (1993), pp. 131-136.
29. A. H. STROUD, Approximate Calculation of Multiple Integrals, Prentice-Hall, Englewood Cliffs, NJ, 1971.
30. V. TCHAKALOFF, Formules de cubatures mécaniques à coefficients non négatifs, Bull. Sci. Math. (2), 81 (1957), pp. 123-134.