Constructing fully symmetric cubature formulae for the sphere

We construct symmetric cubature formulae of degrees in the 13-39 range for the surface measure on the unit sphere. We exploit a recently published correspondence between cubature formulae on the sphere and on the triangle. Specifically, a fully symmetric cubature formula for the surface measure on the unit sphere corresponds to a symmetric cubature formula for the triangle with weight function (u 1 u 2 u 3 ) -1/2 , where u 1 , u 2 , and u 3 are homogeneous coordinates.

[1]  V. Lebedev,et al.  A QUADRATURE FORMULA FOR THE SPHERE OF THE 131ST ALGEBRAIC ORDER OF ACCURACY , 1999 .

[2]  Jan Szynal,et al.  An upper bound for the Laguerre polynomials , 1998 .

[3]  Louis E. Rosier.  A Note on Presburger Arithmetic with Array Segments, Permutation and Equality , 1986, Inf. Process. Lett..

[4]  J. I. Maeztu,et al.  Consistent structures of invariant quadrature rules for the n -simplex , 1995 .

[5]  R. Cools,et al.  Monomial cubature rules since “Stroud”: a compilation , 1993 .

[6]  Patrick Keast,et al.  Fully Symmetric Integration Formulas for the Surface of the Sere in S Dimensions , 1983 .

[7]  V. I. Lebedev,et al.  Spherical quadrature formulas exact to orders 25–29 , 1977 .

[8]  V. I. Lebedev,et al.  Quadratures on a sphere , 1976 .

[9]  P. Keast.  Moderate-degree tetrahedral quadrature formulas , 1986 .

[10]  A. D. McLaren,et al.  Optimal numerical integration on a sphere , 1963 .

[11]  Constructing cubature formulae for spheres and balls , 1999 .

[12]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[13]  Yuan Xu,et al.  Orthogonal polynomials and cubature formulae on spheres and on simplices , 1998 .

[14]  Yuan Xu,et al.  ORTHOGONAL POLYNOMIALS AND CUBATURE FORMULAE ON SPHERES AND ON BALLS , 1998 .

[15]  D. A. Dunavant.  High degree efficient symmetrical Gaussian quadrature rules for the triangle , 1985 .

[16]  Morris Weisfeld,et al.  Orthogonal polynomials in several variables , 1959, Numerische Mathematik.

[17]  H. Engels,et al.  Numerical Quadrature and Cubature , 1980 .

[18]  Patrick Keast.  Cubature formulas for the surface of the sphere , 1987 .

[19]  Ronald Cools,et al.  A survey of numerical cubature over triangles , 1993 .

[20]  James N. Lyness,et al.  Moderate degree symmetric quadrature rules for the triangle j inst maths , 1975 .

[21]  P. Bazant,et al.  Efficient Numerical Integration on the Surface of a Sphere , 1986 .