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为计算机代数设计一个标准和高效的程序库，利用各种技术和技术（SIMD，多核，…）在商用处理器上实现更高的性能。它首先扩展了一直在进行的研究项目及其结果，包括提高效率的奇怪方法和渐近快速算法的使用。本课题的研究成果包括：(1)总结了快速矩阵乘法算法在计算机代数中的行为，并提出了在实际应用中适当的方法。(2)为了实现更高的效率，实验了超线程技术和多核技术在共享内存多线程中的应用。结果表明，共享内存并行处理下的内存管理以及数学函数级别与计算粒度的关系是实现多线程符号计算的关键。(3)针对隐函数的应用，研究了隐函数绘制的昂贵算法，并提出了一种与区间算法相关的新算法。进一步的研究，结合对区间算法和高性能多线程计算的深入理解，可以成为一个新的研究课题。(4)最后一个是易于处理数学表达式的实验软件。新的web浏览器工具采用最新的web技术，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.