SPHERE_LEBEDEV_RULE
Quadrature Rules for the Unit Sphere

SPHERE_LEBEDEV_RULE is a C++ library which computes Lebedev quadrature rules 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 FORTRAN90 version and a MATLAB version.

Related Programs:

ANNULUS_RULE, a C++ library which computes a quadrature rule for estimating integrals of a function over the interior of a circular annulus in 2D.

CIRCLE_RULE, a C++ library which computes quadrature rules over the circumference of the unit 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_CVT, a FORTRAN90 library which creates a mesh of well-separated points using Centroidal Voronoi Tessellations, over the surface of the unit sphere in 3D.

SPHERE_DESIGN_RULE, a FORTRAN90 library which returns point sets on the surface of the unit sphere, known as "designs", which can be useful for estimating integrals on the surface, among other uses.

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 unit sphere.

SPHERE_LEBEDEV_RULE, a dataset directory which contains sets of points on a unit sphere which can be used for quadrature rules of a known precision;

SPHERE_LEBEDEV_RULE_DISPLAY, a MATLAB program which reads a file defining a Lebedev quadrature rule for the unit sphere and displays the point locations.

SPHERE_MONTE_CARLO, a C++ library which applies a Monte Carlo method to estimate the integral of a function 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;

SPHERE_TRIANGLE_QUAD, a C++ library which estimates the integral of a function over a spherical triangle.

SPHERE_XYZ_DISPLAY, a MATLAB program which reads XYZ information defining points in 3D, and displays a unit sphere and the points in the MATLAB 3D graphics window.

SPHERE_XYZ_DISPLAY_OPENGL, a C++ program which reads XYZ information defining points in 3D, and displays a unit sphere and the points, using OpenGL.

SPHERE_XYZF_DISPLAY, a MATLAB program which reads XYZF information defining points and faces, and displays a unit sphere, the points, and the faces, in the MATLAB 3D graphics window. This can be used, for instance, to display Voronoi diagrams or Delaunay triangulations on the unit sphere.

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.