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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Aho, A., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms., Addison-Wesley, Mass 1974
Boas, R.P., Buck, R.C.: Polynomial Expansions of Analytic Functions. Academic Press, New York 1964
Bukhshtaber, V.M., Kholodov, A.N.: Groups of Formal Diffeomorphisms of Superline Generating Functions for Polynomial Sequences and Functional Equations. Isv. AN USSR, Ser. Math.53(3), 944–970 (1989)
Dilts, G.A.: Computation of Spherical Harmonic Expansion Coefficients via FFT's, J. Computat. Phys.57, 349–453 (1985)
Elliott, D.F., Rao, K.R.: Fast Transforms: Algorithms, Analyses, Applications. New York: Academic Press 1982
Erdelyi, A., et al.: Higher Transcedential Functions. The Bateman Manuscript Project. V. 3, New York: McGraw-Hill 1955
Freeden, W.: On Integral Formulae of the (Unit) Sphere and Their Application to Numeric Computation of Integrals., Computing,25, 131–146 (1980)
Freedman, W., Reuter, R.: An Efficient Algorithm for the Generation of Homogeneous Harmonic Polynomials. In: Scientific Software Systems. Mason, J.C., Cox, M. G. (eds.), pp. 166–180 Chapman and Hall 1990
Frumkin, M., Kholodov, A.: O(n log2n) algorithm for the multiplication of the umbral matrix by a vector, Unpublished Manuscript 1990
Ph. R. Gantmacher, The matrix Theory, Chelsia 1967
Grigor'ev, D. Yu.: Additive Complexity in Directed Computations. Theor. Comp. Sci., vol. V.19, pp. 39–67
Muller, C.: Spherical Harmonics, Lecture Notes in Mathematics vol.17, Berlin, Heidelberg New York: Springer 1966
Nussbaumer, H.J.: Fast Fourier Transform and Convolution Algorithms. Berlin, Heildelberg New York: Springer 1982
Press, W.H. et al.: Numercal Recipes. Cambridge Univ. Press 1988.
Rabiner, L.R., Gold, B.: Theory and Application of Digital Signal Processing. Englewood Cliffs, N.J.: Prentice-Hall 1975
Roman, S.: The Umbral Calculus New York: Academic Press 1984
Vilenkin, N. Ja.: Special Functions and the Theory of Group Representations. Moscow, Nauka (in Russian) 1965
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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