SPHERE_LEBEDEV_RULE
Quadrature Rules for the Unit Sphere


SPHERE_LEBEDEV_RULE is a C library which computes a Lebedev quadrature rule over the surface of the unit sphere in 3D.

Vyacheslav Lebedev determined a family of 65 quadrature rules for the unit sphere, increasing in precision from 3 to 131, by 2 each time. This software library computes any one of a subset of 32 of these rules.

Each rule is defined as a list of N values of theta, phi, and w. Here:

Of course, each pair of values (thetai, phii) has a corresponding Cartesian representation:

xi = cos ( thetai ) * sin ( phii )
yi = sin ( thetai ) * sin ( phii )
zi = cos ( phii )
which may be more useful when evaluating integrands.

The integral of a function f(x,y,z) over the surface of the unit sphere can be approximated by

integral f(x,y,z) = 4 * pi * sum ( 1 <= i <= N ) f(xi,yi,zi)

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

SPHERE_LEBEDEV_RULE is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Programs:

CIRCLE_RULE, a C library which computes quadrature rules over the circumference of a circle in 2D.

CUBE_FELIPPA_RULE, a C library which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.

PYRAMID_FELIPPA_RULE, a C library which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.

SPHERE_EXACTNESS, a C program which tests the monomial exactness of a quadrature rule on the surface of the unit sphere in 3D.

SPHERE_GRID, a C library which provides a number of ways of generating grids of points, or of points and lines, or of points and lines and faces, over the surface of the unit sphere in 3D.

SPHERE_LEBEDEV_RULE, a dataset directory which contains quadrature rules over the surface of the unit sphere in 3D.

SPHERE_QUAD, a C library which approximates an integral by applying a triangulation over the surface of the unit sphere in 3D.

SQUARE_FELIPPA_RULE, a C library which returns the points and weights of a Felippa quadrature rule over the interior of a square in 2D.

TETRAHEDRON_FELIPPA_RULE, a C library which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

TRIANGLE_FEKETE_RULE, a C library which defines Fekete rules for interpolation or quadrature over the interior of a triangle in 2D.

TRIANGLE_FELIPPA_RULE, a C library which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.

WEDGE_FELIPPA_RULE, a C library which returns quadratures rules for approximating integrals over the interior of the unit wedge in 3D.

Reference:

  1. Axel Becke,
    A multicenter numerical integration scheme for polyatomic molecules,
    Journal of Chemical Physics,
    Volume 88, Number 4, 15 February 1988, pages 2547-2553.
  2. Vyacheslav Lebedev, Dmitri Laikov,
    A quadrature formula for the sphere of the 131st algebraic order of accuracy,
    Russian Academy of Sciences Doklady Mathematics,
    Volume 59, Number 3, 1999, pages 477-481.
  3. Vyacheslav Lebedev,
    A quadrature formula for the sphere of 59th algebraic order of accuracy,
    Russian Academy of Sciences Doklady Mathematics,
    Volume 50, 1995, pages 283-286.
  4. Vyacheslav Lebedev, A.L. Skorokhodov,
    Quadrature formulas of orders 41, 47, and 53 for the sphere,
    Russian Academy of Sciences Doklady Mathematics,
    Volume 45, 1992, pages 587-592.
  5. Vyacheslav Lebedev,
    Spherical quadrature formulas exact to orders 25-29,
    Siberian Mathematical Journal,
    Volume 18, 1977, pages 99-107.
  6. Vyacheslav Lebedev,
    Quadratures on a sphere,
    Computational Mathematics and Mathematical Physics,
    Volume 16, 1976, pages 10-24.
  7. Vyacheslav Lebedev,
    Values of the nodes and weights of ninth to seventeenth order Gauss-Markov quadrature formulae invariant under the octahedron group with inversion,
    Computational Mathematics and Mathematical Physics,
    Volume 15, 1975, pages 44-51.

Source Code:

Examples and Tests:

List of Routines:

You can go up one level to the C source codes.


Last revised on 13 September 2010.