International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B, ch. 1.3, pp. 5758

The mathematical analysis of the structure of DFT computations has brought to light a broad variety of possibilities for reducing or reshaping their arithmetic complexity. All of them are `analytic' in that they break down large transforms into a succession of smaller ones.
These results may now be considered from the converse `synthetic' viewpoint as providing a list of procedures for assembling them:
The simplest DFT may then be carried out into a global algorithm in many different ways. The diagrams in Fig. 1.3.3.1 illustrate a few of the options available to compute a 400point DFT. They may differ greatly in their arithmetic operation counts.