# sphere_lebedev_rule

sphere_lebedev_rule, a C 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 (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 code which computes a quadrature rule for estimating integrals of a function over the interior of a circular annulus in 2D.

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

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

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

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

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

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

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

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

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

WEDGE_FELIPPA_RULE, a C 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,
Volume 59, Number 3, 1999, pages 477-481.
3. Vyacheslav Lebedev,
A quadrature formula for the sphere of 59th algebraic order of accuracy,
Volume 50, 1995, pages 283-286.
4. Vyacheslav Lebedev, A.L. Skorokhodov,
Quadrature formulas of orders 41, 47, and 53 for the sphere,
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 09 July 2020.