The crystallographic fast Fourier transform. Recursive symmetry reduction

Algorithms are presented for maximally efficient computation of the crystallographic fast Fourier transform (FFT). The approach is applicable to all 230 space groups and allows reduction of both the computation time and the memory usage by a factor equal to the number of symmetry operators. The central idea is a recursive reduction of the problem to a series of transforms on grids with no special points. The maximally efficient FFT for such grids has been described in previous papers by the same authors. The interaction between the grid size factorization and the symmetry operators and its influence on the algorithm design are discussed.