Recursive computation of spherical harmonic rotation coefficients of large degree

Computation of the spherical harmonic rotation coefficients or elements of Wigner’s dmatrix 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 are 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-FriedrichsLewy condition and the von Neumann method for stability of finite-difference schemes for solution of PDEs. A recursive algorithm of minimal complexity O ( n2 ) for degree n and FFTbased algorithms of complexity O ( n2 log n ) 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 . 103 in double precision, while the former algorithm was tested for large n (up to 104 in our experiments) and demonstrated better performance and accuracy compared to the FFT-based algorithm.