Computer Algebra and Supercomputing

The overall subject of this research project is the application of supercomputers to computational algebra。The major research results are described briefly in the following.·A new modular algorithm which uses the Chinese remainder algorithm for sparse multivariate polynomial interpolation is developed，which determines unknown polynomials from their numeric values produced by efficient data-parallel processing on supercomputers。Also，its brief analysis and comparison with other algorithms are done。To evaluate its practical efficiency，the algorithm was fully implemented and tested in REDUCE，and the effort of its parallelization using C and KLIC，was made.·The new algorithm，recently developed by the investigator，for solving a system of algebraic equations was implemented on Risa/Asir，to clarify problems and appropriate methods to be used，in applying vector or parallel processing to the algorithm。It turned out that in the case of large-scale systems of equations，the computing met…More hod considered suffers from a very large amount of data to be transferred between a computer algebra system and a supercomputer each other，and therefore，requires an efficient device for interprocess communication。In this experiment，the Grobner-basis package of Risa/Asir was much improved to become the most efficient package in the world，enabling to perform maximal-scale computations.·Various algorithms for polynomial factorization are investigated theoretically and empirically。The significant findings of the experiment with a supercomputer are the facts that vector processing is substantially effective，and that among others，the Zassenhaus algorithm is generally most efficient and useful。The vectorized programs run so fast that they can factor polynomials of such high degrees as having never been tried，within a reasonable amount of time。Further investigation has been being continued to clarify how high-degree polynomial we can factorize using the state-of-the-art supercomputers。This has led to a deeper understanding of the algorithms，and brought a new algorithm。·The mechanism for network-transparent interprocess communication in Risa/Asir was slightly extended，and it was tested and evaluated very briefly，for its future use or generalization in combining the computer algebra system and supercomputing environments。Less：Less

The overall subject of this research project is the application of supercomputers to computational algebra. The major research results are described briefly in the following.・A new modular algorithm which uses the Chinese remainder algorithm for sparse multivariate polynomial interpolation is developed, which determines unknown polynomials from their numeric values produced by efficient data-parallel processing on supercomputers. Also, its brief analysis and comparison with other algorithms are done. To evaluate its practical efficiency, the algorithm was fully implemented and tested in REDUCE,and the effort of its parallelization using C and KLIC,was made.・The new algorithm, recently developed by the investigator, for solving a system of algebraic equations was implemented on Risa/Asir, to clarify problems and appropriate methods to be used, in applying vector or parallel processing to the algorithm. It turned out that in the case of large-scale systems of equations, the computing met … More hod considered suffers from a very large amount of data to be transferred between a computer algebra system and a supercomputer each other, and therefore, requires an efficient device for interprocess communication. In this experiment, the Grobner-basis package of Risa/Asir was much improved to become the most efficient package in the world, enabling to perform maximal-scale computations.・Various algorithms for polynomial factorization are investigated theoretically and empirically. The significant findings of the experiment with a supercomputer are the facts that vector processing is substantially effective, and that among others, the Zassenhaus algorithm is generally most efficient and useful. The vectorized programs run so fast that they can factor polynomials of such high degrees as having never been tried, within a reasonable amount of time. Further investigation has been being continued to clarify how high-degree polynomial we can factorize using the state-of-the-art supercomputers. This has led to a deeper understanding of the algorithms, and brought a new algorithm.・The mechanism for network-transparent interprocess communication in Risa/Asir was slightly extended, and it was tested and evaluated very briefly, for its future use or generalization in combining the computer algebra system and supercomputing environments. Less