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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研究一种适合实际使用的渐近快速计算机代数算法和开发高效软件的方法，主要侧重于基本操作。更具体地说，我们以向量和矩阵为计算对象，利用各种算法和编程技术，开发出高效的软件。首先，研究了矩阵乘法的strassen型快速算法，阐明了其快速的条件和原因。即使通过计算算术运算的数量来进行详细的分析，在某些情况下显示的令人难以置信的速度也不能完全解释，然而，我们注意到计算时间与空间复杂性密切相关。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代价高得多，而加法运算不会改变表达式的稀疏性，从而改变它们的乘法代价，那么快速算法就会生效。模运算就是一个典型的例子。我们的实验加强了必要性，我们早就意识到，为开发有效的基本线性运算库，如BIAS，模块化算法，这将有各种各样的应用。我们的第二个主题是这个名为MBLAS的软件的开发。我们为线性代数定义了一组类似于BLAS的子程序，并为各种应用设计了适当的接口，包括对密集单变量多项式算法的应用。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。我们已经探索了几种加速技术，例如使用表、用乘法进行除法转换、使用溢出作为可管理的值。此外，我们还尝试了向量处理，使用短向量SIMD指令处理流数据。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。我们得到的另一个知识是，快速矩阵乘法算法不适用于稀疏矩阵。即使是为稀疏矩阵量身定制的矩阵表示也是如此。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效果。适当的矩阵表示的研究已成为我们项目的第三个主题，并将留给未来的研究课题。另一位研究者开发了一些高级算法，并实现了高效的代数计算软件，除了作为首席开发人员维护和改进计算机代数系统Risa/Asir之外。他的工作包括不断改进groebner基包，开发有限域上多元多项式分解的算法和实现，以及动态评估的模块化方法。少