Research for practical use of fast algorithms for computer algebra and software development

计算机代数和软件开发快速算法的实际应用研究

基本信息

批准号：
14580365
负责人：
MURAO Hirokazu
金额：
$ 1.6万
依托单位：
The University of Electro-Communications
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-14580365/
关键词：
computer algebra asymptotically fast algorithm Strassen-Winograd algorithm representation of math formula modular arithmetic BLAS vector operation SIMD operation 高速アルゴリズム行列乗算代数的数の計算 Risa Asir Dynamic Evaluation 基本線形演算有限体の演

项目摘要

The overall subject of this project is the investigation of an appropriate method for practical use of asymptotically fast algorithms for computer algebra and the development of efficient software, with major emphasis upon the basic operations. More concretely, adopting vectors and matrices as a computing target, we developed efficient software, by making use of various algorithms and programming techniques. First, we investigated the Strassen-type fast algorithm for matrix multiplication, to clarify the conditions and the reason of its fastness. Even with detailed analysis done by counting the number of arithmetic operations, the incredible fastness revealed in some cases cannot be explained completely, however, we noticed instead that computing time is closely related with space complexity. As an experience rule, we have learnt that the fast algorithm takes effect if the multiplication of matrix elements is much costlier than the additive operations, and additive operations do not ch … More ange the sparseness of the expressions and therefore their cost for multiplication. Modular arithmetics is a typical example of this kind. Our experiment strengthened the necessity, we have long been aware of, for the development of efficient library of basic linear operations, like BIAS, for modular arithmetics, which would have a wide variety of applications. Our second topic is the development of this software, called MBLAS. We defined a set of subroutines for linear algebra, in analogy with BLAS, and designed an appropriate interface for various applications, including the application to dense univariate polynomial arithmetics. With modular arithmetics, removal of division is a key to speed-up, especially for the case of matrix or vector operations. We have explored several techniques for speedup, such as use of tables, conversionof division by multiplication, use of overflow as a manageable value. Also, we experimented vector processing, using short-vector SIMD instructions for streaming data. Our extensive empirical study indicated that the effect of these techniques heavily depends on the hardware specification. One more knowledge we have obtained is the fact that the fast matrix-multiplication algorithm is not suited for sparse matrices. This is true even with matrix representation tailored for sparse matrices. Throughout this experiment, we noticed that the use of array for vectors and matrices is of almost no effect, as far as symbolic computation is concerned. Investigation of appropriate matrix representation has become the third subject in our project and is left for a topic of future study.Another investigator developed some high-level algorithms and realized efficient software for algebraic computation, besides the maintenance and the improvement of a computer algebra system Risa/Asir as a chief development staff. His work includes the continuous effort for improving the Groebner-basis package, development of algorithm and realization for factorization of multivariate polynomials over finite fields, and modular method for dynamic evaluation. Less

对计算机LGEBRA的实用性UF的合适方法的总体主题，并为基本操作提供了重点，我们通过制造Vargorithms和Programming Technique进行了有效的软件。用于矩阵多莱方算术操作的LGORITHM，在某种情况下，不可思议的牢固性无法解释与太空编译规则密切相关，我们了解到，如果矩阵溶液的矩阵矩阵的成本更高，则快速算法会生效。。我们的第二个主题是Theis软件的开发，称为MBLA，以与Blas类似，为线性代数定义了一组子程序，并为各种应用程序设计了一个模块化算术。 - 尤其是对于矩阵或向量操作的情况，我们还探索了几种加速技术，我们使用短载体SIMD教学来实验矢量处理在他的实验中，矩阵不是稀疏的矩阵，我们注意到载体和基质几乎没有效果。级别的级别。