A fast algorithm for three-dimensional potential fields calculation: fast Fourier transform on multipoles

In this paper, we present a fast algorithm for rapid calculation of the potential fields in three dimensions. This method arises from an observation that potential evaluation using the multipole to local expansion translation operator can be expressed as a series discrete convolutions of the multipole moments with their associated spherical harmonics functions. The high efficiency of the algorithm is primarily due to the use of FFT algorithms to evaluate the numerous discrete convolutions. We refer to it as the Fast Fourier Transform on Multipoles (FFTM) method. It is demonstrated that FFTM is an accurate method. It is significantly more accurate than FMM for a given order of expansion. It is also shown that the algorithm has computational complexity of O(Na), where a ranges from 1.0 to 1.3.