The crystallographic fast Fourier transform. IV. FFT-asymmetric units in the reciprocal space

New algorithms have been outlined for efficient calculation of the fast Fourier transform of data revealing crystallographic symmetries in previous papers by Rowicka, Kudlicki & Otwinowski [Acta Cryst. (2002), A58, 574-579; Acta Cryst. (2003), A59, 172-182; Acta Cryst. (2003), A59, 183-192]. The present paper deals with three implementation-related issues, which have not been discussed before. First, the shape of the FFT-asymmetric unit in the reciprocal space is discussed in detail. Next, a method is presented of reducing symmetry in-place, without the need to allocate memory for intermediate results. Finally, there is a discussion on how the algorithm can be used for the inverse Fourier transform. The results are derived for the case of the one-step symmetry reduction [Rowicka, Kudlicki & Otwinowski (2003). Acta Cryst. A59, 172-182]. The algorithms are also an important step in the more complicated cases of centered lattices [Rowicka, Kudlicki & Otwinowski (2003). Acta Cryst. A59, 183-192] and space groups with non-removable special positions, such as cubic groups [Rowicka, Kudlicki & Otwinowski (2004), in preparation]. In the present paper, as in our previous ones, complex-to-complex FFTs only are dealt with. Modifications needed to adapt the results to data with Hermitian symmetry will be described in our forthcoming article [Kudlicki, Rowicka & Otwinowski (2004), in preparation].