sphere_lebedev_rule

sphere_lebedev_rule, a Fortran77 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:

• theta is a longitudinal angle, measured in degrees, and ranging from -180 to +180.
• phi is a latitudinal angle, measured in degrees, and ranging from 0 to 180.
• w is a weight.

Of course, each pair of values (theta_i, phi_i) has a corresponding Cartesian representation:

x_i = cos ( theta_i ) * sin ( phi_i )
y_i = sin ( theta_i ) * sin ( phi_i )
z_i = cos ( phi_i )

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(x_i,y_i,z_i)

Licensing:

The information on this web page is 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

f77_rule, a Fortran77 code which computes a quadrature rule which estimates the integral of a function f(x), which might be defined over a one dimensional region (a line) or more complex shapes such as a circle, a triangle, a quadrilateral, a polygon, or a higher dimensional region, and which might include an associated weight function w(x).

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:

• sphere_lebedev_rule.f, the source code;
• sphere_lebedev_rule.sh, commands to compile the source code;