sphere_lebedev_rule
sphere_lebedev_rule,
a MATLAB 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} )
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 MIT 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 MATLAB code which
computes a quadrature rule for estimating integrals of a function
over the interior of a circular annulus in 2d.
cube_felippa_rule,
a MATLAB code which
returns the points and weights of a felippa quadrature rule
over the interior of a cube in 3d.
pyramid_felippa_rule,
a MATLAB code which
returns felippa's quadratures rules for approximating integrals
over the interior of a pyramid in 3d.
sphere_cvt,
a MATLAB code which
creates a mesh of wellseparated 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_grid,
a MATLAB 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 sphere which can be used for
quadrature rules of a known precision;
sphere_lebedev_rule_test
sphere_lebedev_rule_display,
a MATLAB code which
reads a file defining a lebedev quadrature rule for the sphere and
displays the point locations.
sphere_quad,
a MATLAB code which
approximates an integral over the surface of the unit sphere
by applying a triangulation to the surface;
sphere_voronoi,
a MATLAB code which
computes and plots the voronoi diagram of points
over the surface of the unit sphere in 3d.
square_felippa_rule,
a MATLAB code which
returns the points and weights of a felippa quadrature rule
over the interior of a square in 2d.
tetrahedron_felippa_rule,
a MATLAB code which
returns felippa's quadratures rules for approximating integrals
over the interior of a tetrahedron in 3d.
triangle_fekete_rule,
a MATLAB code which
defines fekete rules for quadrature or interpolation
over the interior of a triangle in 2d.
triangle_felippa_rule,
a MATLAB code which
returns felippa's quadratures rules for approximating integrals
over the interior of a triangle in 2d.
wedge_felippa_rule,
a MATLAB code 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:

available_table.m,
returns the availability of a Lebedev rule.

gen_oh.m,
generates points under OH symmetry.

ld_by_order.m,
returns a Lebedev angular grid given its order.

ld0006.m,
computes the 6 point Lebedev angular grid.

ld0014.m,
computes the 14 point Lebedev angular grid.

ld0026.m,
computes the 26 point Lebedev angular grid.

ld0038.m,
computes the 38 point Lebedev angular grid.

ld0050.m,
computes the 50 point Lebedev angular grid.

ld0074.m,
computes the 74 point Lebedev angular grid.

ld0086.m,
computes the 86 point Lebedev angular grid.

ld0110.m,
computes the 110 point Lebedev angular grid.

ld0146.m,
computes the 146 point Lebedev angular grid.

ld0170.m,
computes the 170 point Lebedev angular grid.

ld0194.m,
computes the 194 point Lebedev angular grid.

ld0230.m,
computes the 230 point Lebedev angular grid.

ld0266.m,
computes the 266 point Lebedev angular grid.

ld0302.m,
computes the 302 point Lebedev angular grid.

ld0350.m,
computes the 350 point Lebedev angular grid.

ld0434.m,
computes the 434 point Lebedev angular grid.

ld0590.m,
computes the 590 point Lebedev angular grid.

ld0770.m,
computes the 770 point Lebedev angular grid.

ld0974.m,
computes the 974 point Lebedev angular grid.

ld1202.m,
computes the 1202 point Lebedev angular grid.

ld1454.m,
computes the 1454 point Lebedev angular grid.

ld1730.m,
computes the 1730 point Lebedev angular grid.

ld2030.m,
computes the 2030 point Lebedev angular grid.

ld2354.m,
computes the 2354 point Lebedev angular grid.

ld2702.m,
computes the 2702 point Lebedev angular grid.

ld3074.m,
computes the 3074 point Lebedev angular grid.

ld3470.m,
computes the 3470 point Lebedev angular grid.

ld3890.m,
computes the 3890 point Lebedev angular grid.

ld4334.m,
computes the 4334 point Lebedev angular grid.

ld4802.m,
computes the 4802 point Lebedev angular grid.

ld5294.m,
computes the 5294 point Lebedev angular grid.

ld5810.m,
computes the 5810 point Lebedev angular grid.

order_table.m,
returns the order of a Lebedev rule.

precision_table.m,
returns the precision of a Lebedev rule.

xyz_to_tp.m,
converts (X,Y,Z) to (Theta,Phi) coordinates on the unit sphere.
Last revised on 30 January 2019.