Abstract
Algorithms for expansion over spherical harmonics are often used in electrostatic field calculation, calculation of the density functions in quantum chemistry and calculation of molecular surfaces. It usually includes expansion over spherical harmonics of degrees to several dozens. The usual method is to use an integration method over some grid on the unit sphere and in fact is a multiplication of the matrix of values of spherical harmonics in the grid points by a vector of values of the expanding function in the set of points. This algorithm executes O(NL2) operations whereN is the number of the grid points andL is the maximal degree of the spherical harmonics involved. We provide an algorithm of complexity O(NLlog2 L) for multiplication of the matrix of values of spherical harmonics in points of an arbitrary grid on the unit sphere. The algorithm is based on interrelation between spherical harmonics and Legendre polynomials and on a fast algorithm for expansion over Legendre polynomials.
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Frumkin, M. A fast algorithm for expansion over spherical harmonics. AAECC 6, 333–343 (1995). https://doi.org/10.1007/BF01198013
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DOI: https://doi.org/10.1007/BF01198013