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

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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.