Design and Implementation of a Parallel Prime Edge-Length Symmetric FFT

Abstract

A symmetric fast Fourier transform, or Symmetric FFT, is a discrete Fourier transform (DFT) algorithm designed to take advantage of known, well-structured repetitions in the input data, to speed up computations. These repetitions are usually called symmetries. The solution of some large scale problems, such as atomic structure configurations or Volterra filtering, depend on the existence of highly optimized symmetric FFTs. The design and implementation of symmetric FFTs brings about several problems of its own. Multidimensional composite edge-length symmetric FFTs computing the DFT in a set of M nonredundant data points in O(M log M) time, are currently known. No similar bound is known for the time complexity of prime edge-length symmetric FFT methods. Indeed, current prime edge-length symmetric FFT methods yield polynomial complexity bounds, and therefore, are not worth considering for large scale problems. This paper develops and tests a parallel multidimensional symmetric FFT of prime edge-length, whose performance is significantly better than that of the usual FFT for large scale problems.