A Parallel Prime Edge-Length Crystallographic FFT

Abstract

Although other methods are available, computational X-ray crystallography is still the most accurate way of determining the atomic structure of crystals. For large scale problems such as protein or virus structure determination, a huge amount of three-dimensional discrete Fourier transforms (DFT) conform the core computation in these methods. Despite the fact that highly efficient fast Fourier transform (FFT) implementations are available, significant improvements can be obtained by using FFT variants tailored to crystal structure calculations. These variants, or crystallographic FFTs, use a-priori knowledge of the specimen's crystal symmetries to lower the operation count and storage requirement of a usual, asymmetric FFT. The design and implementation of crystallographic FFTs brings about several problems of its own. And, as is usually the case with prime length FFTs, prime edge-length crystallographic FFTs pose the hardest challenges among them. This paper develops and tests a parallel multidimensional crystallographic FFT of prime edge-length, whose performance is significantly better than that of the usual FFT.