Fast Spherical Harmonic Transform Algorithm based on Generalized Fast Multipole Method

Spherical harmonic transform is the most important orthogonal function transform only except Fourier transform, and is used not only for climate simulation and signal processing but also for a base of several numerical algorithms. Fast Fourier Transform (FFT), which runs in time O(N logN) is quite well known, but, for spherical harmonic transform, there is no fast algorithm which is as simple as FFT. We have proposed a fast transform algorithm for spherical harmonic transform using Fast Multipole Method (FMM)[17, 25], and its program is publicly available as FLTSS[24, 20]. Our algorithm computes the transform using polynomial interpolations, and to make it possible, we have introduced split Legendre functions[25], which nearly double the computational costs, and also affect the numerical stability[21]. To reduce computational costs, we tried fast transform without using split Legendre functions[22, 23] with the help of generalized FMM[29, 28]. This algorithm (we call this new algorithm and the former old algorithm) runs faster than the old algorithm with improved numerical stability. This paper discusses some details of the new algorithm and its implementation.