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:
Of course, each pair of values (theta_{i}, phi_{i}) has a corresponding Cartesian representation:
x_{i} = cos ( theta_{i} ) * sin ( phi_{i} )which may be more useful when evaluating integrands.
y_{i} = sin ( theta_{i} ) * sin ( phi_{i} )
z_{i} = cos ( phi_{i} )
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})
The computer code and data files described and made available on this web page are distributed under the MIT license
sphere_lebedev_rule is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.
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