The crystallographic fast Fourier transform. IV. FFTasymmetric 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].