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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

这个项目的总体主题是调查一个适当的方法，实际使用的渐近快速算法的计算机代数和有效的软件开发，主要强调基本操作。更具体地说，采用向量和矩阵作为计算目标，我们开发了高效的软件，利用各种算法和编程技术。首先，我们研究了Strassen型矩阵乘法快速算法，阐明了其快速性的条件和原因。即使通过计算算术运算的数量来进行详细的分析，在某些情况下所揭示的令人难以置信的快速性也无法完全解释，然而，我们注意到计算时间与空间复杂度密切相关。作为经验法则，我们已经了解到，如果矩阵元素的乘法比加法运算的开销大得多，并且加法运算不需要矩阵元素的乘法，则快速算法有效。 ...更多信息改变表达式的稀疏性，从而改变它们的乘法成本。模算术是这类运算的一个典型例子。我们的实验加强了必要性，我们早就意识到，开发有效的基本线性运算库，如BIAS，用于模块运算，这将有各种各样的应用。我们的第二个主题是这个软件的开发，称为MBLAS。我们定义了一组线性代数的子程序，类似于BLAS，并设计了一个适当的接口，为各种应用程序，包括应用到密集的一元多项式算术。对于模运算，去除除法是加速的关键，特别是对于矩阵或向量运算的情况。我们已经探索了几种加速技术，如使用表，除法与乘法的转换，使用溢出作为一个可管理的值。此外，我们还试验了向量处理，使用短向量SIMD指令进行流数据处理。我们广泛的实证研究表明，这些技术的效果在很大程度上取决于硬件规格。我们得到的另一个认识是，快速矩阵乘算法不适用于稀疏矩阵。这是真的，即使是为稀疏矩阵定制的矩阵表示。在整个实验中，我们注意到，就符号计算而言，对向量和矩阵使用数组几乎没有影响。适当的矩阵表示的调查已成为我们的项目的第三个课题，是留给未来的研究课题。另一名研究员开发了一些高级算法，实现了有效的软件代数计算，除了维护和改进的计算机代数系统Risa/Asir作为首席开发人员。他的工作包括不断努力改进Groebner基包，开发算法和实现有限域上多元多项式的因式分解，以及动态评估的模块化方法。少