Numerically stable algorithms for the computation of reduced unit cells
Numerically stable algorithms for the computation of reduced unit cells
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DOI：
10.1107/s010876730302186x
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发表时间：
2004-01-01
期刊：
影响因子：
--
通讯作者：
Adams, PD
中科院分区：
文献类型：
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作者：
Grosse-Kunstleve, RW;Sauter, NK;Adams, PD
简化晶胞的计算是许多晶体学应用的重要构建模块，但不幸的是，很容易证明晶胞简化算法的传统实现在数值上不稳定。 Krivy & Gruber 的 Niggli-reduction 算法的数值稳定实现 [Acta Cryst。 ( 1976), A32, 297 - 298] 提出。通过在所有旋转点比较中一致使用公差来实现稳定性。公差必须大于累积的舍入误差。还提出了第二种稳定算法，即最小减少，不需要使用容差。它产生具有最小长度和所有角度为锐角或钝角的细胞。该算法是 Gruber 的 Buerger-reduction 算法的简化和修改版本 [Acta Cryst. （1973），A29，433 - 440]。两种算法都经过增强，可以生成基础变化矩阵以及简化单元的参数。
The computation of reduced unit cells is an important building block for a number of crystallographic applications, but unfortunately it is very easy to demonstrate that the conventional implementation of cell reduction algorithms is not numerically stable. A numerically stable implementation of the Niggli-reduction algorithm of Krivy & Gruber [ Acta Cryst. ( 1976), A32, 297 - 298] is presented. The stability is achieved by consistently using a tolerance in all roating- point comparisons. The tolerance must be greater than the accumulated rounding errors. A second stable algorithm is also presented, the minimum reduction, that does not require using a tolerance. It produces a cell with minimum lengths and all angles acute or obtuse. The algorithm is a simplified and modified version of the Buerger-reduction algorithm of Gruber [Acta Cryst. (1973), A29, 433 - 440]. Both algorithms have been enhanced to generate a change-of-basis matrix along with the parameters of the reduced cell.