Mathematical Sciences: Algorithms and Programs for Fourier Transforms and Crystallography

这项探索性研究的小额赠款将支持 数学研究的一般数学结构快速 傅立叶变换算法。 重点将放在计算上 一般情况下的离散傅里叶变换。 对具有晶体学对称性的数据的应用将 被制造。 对相关快速卷积算法的研究， 也可以做。 数据转换和置换算法将 制作。这些是在阶段之间转换数据所必需的 傅立叶变换的数据，并减少和扩大数据， 输入和输出。 算法将适应机器架构 包括向量、并行和分级存储器 机械. 他们将考虑数据流，数据排列 和大规模计算中的数据访问。 群论方法将发挥重要作用， 这种努力。 包括在工作的目标是一个系统的 所有晶体群对称性的分类。 算法和程序的其他部分， 晶体学计算，特别是 问题，将被生产。 这项工作的实施在一个大的一般 晶体学程序包和报告的效率是 计划好了

This Small Grant for Exploratory Research will support mathematical research on general mathematical structures for fast Fourier transforms algorithms. The focus will be on computing the discrete Fourier transform for the general case. Applications to data possessing crystallographic symmetries will be made. Studies on related fast convolution algorithms will also be made. Data transformation and permutation algorithms will be produced. These be necessary for transforming data between stages of the Fourier transforms and for reducing and expanding data for input and output. Algorithms will be adapted to machine architectures including those of vector, parallel and hierarchical memory machines. They will take account of data flow, data permutations and data access in large scale computations. Group theoretical methods will play an important part in this effort. Included in the work is the goal of a systematic classification of the symmetries of all crystallographic groups. Algorithms and programs for others parts of the crystallographic calculation, particularly the refinement problem, will be produced. Implementation of this work in one large general crystallographic program package and report on the efficiency is planned.