SPHERE_LEBEDEV_RULE
Quadrature Rules for the Unit Sphere
SPHERE_LEBEDEV_RULE
is a C 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
(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} )
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(x_{i},y_{i},z_{i})
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 library which
computes a quadrature rule for estimating integrals of a function
over the interior of a circular annulus in 2D.
CIRCLE_RULE,
a C library which
computes quadrature rules
over the circumference of a circle in 2D.
CUBE_FELIPPA_RULE,
a C library which
returns the points and weights of a Felippa quadrature rule
over the interior of a cube in 3D.
PYRAMID_FELIPPA_RULE,
a C library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a pyramid in 3D.
SPHERE_EXACTNESS,
a C program which
tests the monomial exactness of a quadrature rule
on 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 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 library which
approximates an integral by applying a triangulation
over the surface of the unit sphere in 3D.
SQUARE_FELIPPA_RULE,
a C library which
returns the points and weights of a Felippa quadrature rule
over the interior of a square in 2D.
TETRAHEDRON_FELIPPA_RULE,
a C library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a tetrahedron in 3D.
TRIANGLE_FEKETE_RULE,
a C library which
defines Fekete rules for interpolation or quadrature
over the interior of a triangle in 2D.
TRIANGLE_FELIPPA_RULE,
a C library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a triangle in 2D.
WEDGE_FELIPPA_RULE,
a C library which
returns quadratures rules for approximating integrals
over the interior of the unit wedge in 3D.
Reference:

Axel Becke,
A multicenter numerical integration scheme for polyatomic molecules,
Journal of Chemical Physics,
Volume 88, Number 4, 15 February 1988, pages 25472553.

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 477481.

Vyacheslav Lebedev,
A quadrature formula for the sphere of 59th algebraic
order of accuracy,
Russian Academy of Sciences Doklady Mathematics,
Volume 50, 1995, pages 283286.

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 587592.

Vyacheslav Lebedev,
Spherical quadrature formulas exact to orders 2529,
Siberian Mathematical Journal,
Volume 18, 1977, pages 99107.

Vyacheslav Lebedev,
Quadratures on a sphere,
Computational Mathematics and Mathematical Physics,
Volume 16, 1976, pages 1024.

Vyacheslav Lebedev,
Values of the nodes and weights of ninth to seventeenth
order GaussMarkov quadrature formulae invariant under the
octahedron group with inversion,
Computational Mathematics and Mathematical Physics,
Volume 15, 1975, pages 4451.
Source Code:
Examples and Tests:
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 the current YMDHMS date as a time stamp.

XYZ_TO_TP converts (X,Y,Z) to (Theta,Phi) coordinates on the unit sphere.
You can go up one level to
the C source codes.
Last revised on 13 September 2010.