Design of standard program library for computer algebra, and research on its efficient implementation and its applications

本研究项目的目标是利用各种技术和技术(单指令多核、多核、…)，设计一个标准的、高效的计算机代数程序库在商用处理器上获得更高的性能。它首先扩展了正在进行的研究项目及其结果，包括奇怪的效率方法和渐近快速算法的使用。该项目的研究成果包括以下几个方面。(1)总结了快速矩阵乘法算法在计算机代数中的表现，并提出了一种适合于实际应用的方法。(2)为了获得更高的效率，尝试了超线程技术和多核技术在共享内存的多线程中的应用。结果表明，共享内存并行处理下的内存管理以及数学函数层次与计算粒度的关系是多线程符号计算的关键。(3)从应用的角度出发，研究了代价较高的隐函数作图算法，提出了一种与区间运算相关的新算法。进一步的研究，结合对区间算术的深入理解和高性能的多线程计算，可以成为一个新的研究课题。(4)数学公式简易处理实验软件。新的网络浏览器工具采用了最新的网络技术，并且是独一无二的。

The target of this research project is to design a standard and efficient program library for computer algebra, exploiting various techniques and technologies(SIMD, multi-core, …) for higher-performance on the commercially available processors. It started by extending the has-beein-ongoing research project and its results, including weird approach for efficiency and the use of asymptotically fast algorithms.The research results of this project include the following. (1) Summary of the behavior of fast matrix multiplication algorithm in computer algebra, and proposition of an appropriate way of its use in practice. (2) To achieve higher efficiency, the application of hyper-threading technology and multicore technology via multi-threading with shared-memory is experimented. It turned out that the memory management under shared-memory parallel processing and the relation of the levels of mathematical functionalities and the grain size of computations are the keys for multi-threaded symbolic computation. (3) For the application, we investigated the costly algorithm for plotting implicit functions and proposed a new algorithm, relating with interval arithmetics. Further investigation, in conjunction with deep understanding of interval arithmetics and with high-performance multi-thread computations can be a new research subject. (4) The last one is the experimental software for easy handling of mathematical expressions. The new Web-browser tool enjoys the latest Web-technology, and is unique.