Abstract
Computation of the spherical harmonic rotation coefficients or elements of Wigner’s d-matrix is important in a number of quantum mechanics and mathematical physics applications. Particularly, this is important for the fast multipole methods in three dimensions for the Helmholtz, Laplace, and related equations, if rotation-based decomposition of translation operators is used. In these and related problems related to representation of functions on a sphere via spherical harmonic expansions computation of the rotation coefficients of large degree n (of the order of thousands and more) may be necessary. Existing algorithms for their computation, based on recursions, are usually unstable, and do not extend to n. We develop a new recursion and study its behavior for large degrees, via computational and asymptotic analyses. Stability of this recursion was studied based on a novel application of the Courant-Friedrichs-Lewy condition and the von Neumann method for stability of finite-difference schemes for solution of PDEs. A recursive algorithm of minimal complexity \(O(n^{2})\) for degree n and FFT-based algorithms of complexity \(O\left( n^{2}\log n\right)\) suitable for computation of rotation coefficients of large degrees are proposed, studied numerically, and cross-validated. It is shown that the latter algorithm can be used for \(n \lesssim 10^{3}\) in double precision, while the former algorithm was tested for large n (up to \(10^{4}\) in our experiments) and demonstrated better performance and accuracy compared to the FFT-based algorithm.
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Gumerov, N., Duraiswami, R. (2015). Recursive Computation of Spherical Harmonic Rotation Coefficients of Large Degree. In: Balan, R., Begué, M., Benedetto, J., Czaja, W., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 3. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-13230-3_5
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DOI: https://doi.org/10.1007/978-3-319-13230-3_5
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