# sphere_lebedev_rule

sphere_lebedev_rule, a C++ code 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:

• 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)

### Languages:

sphere_lebedev_rule is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Programs:

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CIRCLE_RULE, a C++ code which computes quadrature rules over the circumference of the unit 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_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++ 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 unit sphere which can be used for quadrature rules of a known precision;

SPHERE_MONTE_CARLO, a C++ code which applies a Monte Carlo method to estimate the integral of a function over the surface of the unit sphere in 3D;

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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,