sphere_lebedev_rule

sphere_lebedev_rule, an Octave code 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 MIT license

Languages:

sphere_lebedev_rule is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version.

Related Programs:

sphere_lebedev_rule_test

alpert_rule, an Octave code which sets up an Alpert quadrature rule for functions which are regular, log(x) singular, or 1/sqrt(x) singular.

annulus_rule, an Octave code which computes a quadrature rule for estimating integrals of a function over the interior of a circular annulus in 2d.

cube_felippa_rule, an Octave code which returns the points and weights of a felippa quadrature rule over the interior of a cube in 3d.

pyramid_felippa_rule, an Octave code which returns felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3d.

sphere_cvt, an Octave code 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, an Octave code 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 sphere which can be used for quadrature rules of a known precision;

sphere_lebedev_rule_display, an Octave code which reads a file defining a lebedev quadrature rule for the sphere and displays the point locations.

sphere_quad, an Octave code which approximates an integral over the surface of the unit sphere by applying a triangulation to the surface;

sphere_voronoi, an Octave code which computes and plots the voronoi diagram of points over the surface of the unit sphere in 3d.

square_felippa_rule, an Octave code which returns the points and weights of a felippa quadrature rule over the interior of a square in 2d.

tetrahedron_felippa_rule, an Octave code which returns felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3d.

triangle_fekete_rule, an Octave code which defines fekete rules for quadrature or interpolation over the interior of a triangle in 2d.

triangle_felippa_rule, an Octave code which returns felippa's quadratures rules for approximating integrals over the interior of a triangle in 2d.

wedge_felippa_rule, an Octave code 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:

Last revised on 30 January 2019.