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A fast spherical harmonics transform algorithm

Authors: Reiji Suda and Masayasu Takami
Journal: Math. Comp. 71 (2002), 703-715
MSC (2000): Primary 65T99, 42C10
Published electronically: November 28, 2001
MathSciNet review: 1885622
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Abstract: The spectral method with discrete spherical harmonics transform plays an important role in many applications. In spite of its advantages, the spherical harmonics transform has a drawback of high computational complexity, which is determined by that of the associated Legendre transform, and the direct computation requires time of $O(N^3)$ for cut-off frequency $N$. In this paper, we propose a fast approximate algorithm for the associated Legendre transform. Our algorithm evaluates the transform by means of polynomial interpolation accelerated by the Fast Multipole Method (FMM). The divide-and-conquer approach with split Legendre functions gives computational complexity $O(N^2 \log N)$. Experimental results show that our algorithm is stable and is faster than the direct computation for $N \ge 511$.

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Additional Information

Reiji Suda
Affiliation: Department of Computational Science and Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8603, Japan

Masayasu Takami
Affiliation: Department of Computational Science and Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8603, Japan

Keywords: Spherical harmonics transform, associated Legendre transform, fast transform algorithm, computational complexity
Received by editor(s): January 24, 2000
Received by editor(s) in revised form: July 10, 2000
Published electronically: November 28, 2001
Additional Notes: This research is partly supported by the Japan Society for Promotion of Science (Computational Science and Engineering for Global Scale Flow Systems Project), Grant-in-Aid #11450038 of the Ministry of Education, and the Toyota Physical and Chemical Research Institute.
Article copyright: © Copyright 2001 American Mathematical Society

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