Kudlicki Andrzej, Rowicka Maga, Otwinowski Zbyszek
Department of Biochemistry, UT Southwestern Medical Center at Dallas, 5323 Harry Hines Boulevard, Dallas, TX 75390-9038, USA.
Acta Crystallogr A. 2004 Mar;60(Pt 2):146-52. doi: 10.1107/S0108767303029659. Epub 2004 Feb 17.
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].
在罗维茨卡、库德利茨基和奥特维诺夫斯基之前的论文中[《晶体学报》(2002年)，A58卷，574 - 579页；《晶体学报》(2003年)，A59卷，172 - 182页；《晶体学报》(2003年)，A59卷，183 - 192页]，已经概述了用于高效计算揭示晶体学对称性的数据的快速傅里叶变换的新算法。本文讨论了三个与实现相关的问题，这些问题以前尚未讨论过。首先，详细讨论了倒易空间中FFT不对称单元的形状。其次，提出了一种就地降低对称性的方法，无需为中间结果分配内存。最后，讨论了该算法如何用于逆傅里叶变换。结果是针对一步对称性降低的情况得出的[罗维茨卡、库德利茨基和奥特维诺夫斯基(2003年)。《晶体学报》A59卷，172 - 182页]。这些算法也是处理更复杂的带心晶格情况[罗维茨卡、库德利茨基和奥特维诺夫斯基(2003年)。《晶体学报》A59卷，183 - 192页]以及具有不可去除特殊位置的空间群（如立方群）[罗维茨卡、库德利茨基和奥特维诺夫斯基(2004年)，正在准备中]的重要一步。在本文中，与我们之前的论文一样，只处理复到复的FFT。将在我们即将发表的文章中描述为使结果适用于具有厄米对称性的数据所需的修改[库德利茨基、罗维茨卡和奥特维诺夫斯基(2004年)，正在准备中]。
