# SPHERE_LEBEDEV_RULE Quadrature Rules for the Unit Sphere

SPHERE_LEBEDEV_RULE is a FORTRAN77 library 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)

### Languages:

SPHERE_LEBEDEV_RULE is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

### Related Programs:

CUBE_FELIPPA_RULE, a FORTRAN77 library which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.

PYRAMID_FELIPPA_RULE, a FORTRAN77 library which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.

PYRAMID_RULE, a FORTRAN77 program which computes a quadrature rule over the interior of the unit 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_EXACTNESS, a FORTRAN77 program which tests the monomial exactness of a quadrature rule over the surface of the unit sphere in 3D.

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_INTEGRALS, a FORTRAN77 library which defines test functions for integration over the surface of the unit sphere in 3D.

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

SPHERE_QUAD, a FORTRAN77 library which approximates an integral over the surface of the unit sphere by applying a triangulation to the surface;

SPHERE_VORONOI, a FORTRAN90 program which computes and plots the Voronoi diagram of points over the surface of the unit sphere in 3D.

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 FORTRAN77 library which returns the points and weights of a Felippa quadrature rule over the interior of a square in 2D.

TETRAHEDRON_FELIPPA_RULE, a FORTRAN77 library which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

TRIANGLE_FELIPPA_RULE, a FORTRAN77 library which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.

WEDGE_FELIPPA_RULE, a FORTRAN77 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,
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,
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.

### List of Routines:

• AVAILABLE_TABLE returns the availability of a Lebedev rule.
• GEN_OH generates points under OH symmetry.
• LD_BY_ORDER returns a Lebedev angular grid given its order.
• LD0006 computes the 6 point Lebedev angular grid.
• LD0014 computes the 14 point Lebedev angular grid.
• LD0026 computes the 26 point Lebedev angular grid.
• LD0038 computes the 38 point Lebedev angular grid.
• LD0050 computes the 50 point Lebedev angular grid.
• LD0074 computes the 74 point Lebedev angular grid.
• LD0086 computes the 86 point Lebedev angular grid.
• LD0110 computes the 110 point Lebedev angular grid.
• LD0146 computes the 146 point Lebedev angular grid.
• LD0170 computes the 170 point Lebedev angular grid.
• LD0194 computes the 194 point Lebedev angular grid.
• LD0230 computes the 230 point Lebedev angular grid.
• LD0266 computes the 266 point Lebedev angular grid.
• LD0302 computes the 302 point Lebedev angular grid.
• LD0350 computes the 350 point Lebedev angular grid.
• LD0434 computes the 434 point Lebedev angular grid.
• LD0590 computes the 590 point Lebedev angular grid.
• LD0770 computes the 770 point Lebedev angular grid.
• LD0974 computes the 974 point Lebedev angular grid.
• LD1202 computes the 1202 point Lebedev angular grid.
• LD1454 computes the 1454 point Lebedev angular grid.
• LD1730 computes the 1730 point Lebedev angular grid.
• LD2030 computes the 2030 point Lebedev angular grid.
• LD2354 computes the 2354 point Lebedev angular grid.
• LD2702 computes the 2702 point Lebedev angular grid.
• LD3074 computes the 3074 point Lebedev angular grid.
• LD3470 computes the 3470 point Lebedev angular grid.
• LD3890 computes the 3890 point Lebedev angular grid.
• LD4334 computes the 4334 point Lebedev angular grid.
• LD4802 computes the 4802 point Lebedev angular grid.
• LD5294 computes the 5294 point Lebedev angular grid.
• LD5810 computes the 5810 point Lebedev angular grid.
• ORDER_TABLE returns the order of a Lebedev rule.
• PRECISION_TABLE returns the precision of a Lebedev rule.
• TIMESTAMP prints out the current YMDHMS date as a timestamp.
• XYZ_TO_TP converts (X,Y,Z) to (Theta,Phi) coordinates on the unit sphere.

You can go up one level to the FORTRAN77 source codes.

Last revised on 12 September 2010.