Recursive computation of spherical harmonic rotation coefficients of large degree
| dc.contributor.author | Gumerov, Nail A. | |
| dc.contributor.author | Duraiswami, Ramani | |
| dc.date.accessioned | 2014-03-30T15:44:44Z | |
| dc.date.available | 2014-03-30T15:44:44Z | |
| dc.date.issued | 2014-03-28 | |
| dc.description.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 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-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(n^2 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 <~ 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. | en_US |
| dc.identifier.uri | http://hdl.handle.net/1903/15013 | |
| dc.language.iso | en_US | en_US |
| dc.relation.ispartofseries | UM Computer Science Department;CS-TR-5037 | |
| dc.relation.ispartofseries | UMIACS;UMIACS-TR-2014-04 | |
| dc.title | Recursive computation of spherical harmonic rotation coefficients of large degree | en_US |
| dc.type | Technical Report | en_US |