| Research Abstract |
The overall subject of this research project is the application of vector and/or parallel processing to computational algebra. It is expected to enable us, by making full use of the superior computing power and rich resources of high-performance computers in symbolic and algebraic computation, to perform ultimate-scale symbolic calculations in practice. The following lists the brief descriptions of the major research results.
1. The latest algorithms for distinct degree factorization of polynomials over finite fields, which can be regarded as a break-down method for large-scale problems of polynomials with very high degrees, and their complexities are investigated very deeply. As its result, a simple method for improvement is found. Furthermore, by noticing the obvious algebraic independences in the algorithms, the investigator has developed a new algorithm for parallel processing, taking account of communication latency as well. The result of this work is presented at ISSAC'96 and publ
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iched in the proceedings.
2. The new algorithm for sparse multivariate polynomial interpolation, developed by the investigator a few years ago, was revisited and examined to make its description and analysis more complete. The paper of this work was published in the special issue on parallel symbolic computation of Journal of Symbolic Computation.
3. In recent yers, research on the classical problem of integer GCD calculation has been activated, aiming mainly at its parallel processing, and a few new algorithms are published. In this project, the investigator implemented some of those algorithms in Risa/Asir, to verify their much-improved performance. This implementation is further applied to the Grobner basis calculation for solving a system of algebraic equations, and we observed 6-8% reduction of computing time for solving a realistic large-scale problem. Furthermore, the research group of Risa/Asir has succeeded in computing a very large scale problem related with Grobner basis calculation, which have never veen completed till then in the world, using our new implementation and their thchnology of distributed processing. In their paper, it is reported that out new implementation, along with their idea for removal of integer contents from the intermediate polynomials, contributed very much to the success.
4. Nowadays, it has been becoming a common recognition that the use of the asymptotically fast algorithms for polynomial operations is practical and necessary when treating large scale polynomials. In this project, some of the algorithms are implemented empirically, to be used in the calculations listed here. Through this development and empirical study, the investigator has learned various implementation techniques for attaining practical efficiency.
5. By optimizing the exiting asymptotically fast algorithm, the investigator developed a new fast algorithm for polynomial multipoint evaluation, which is used in various polynomial calculations such as distinct degree factorization. At the same time, its parallelization, with communication latency being taken into account, is considered. A paper on this work is submitted and accepted for presentation at PASCO'97, and its implementation in KLIC for parallel processing is in progress.
As for distributed and cooperative processing, the minimal but sufficient functions of Risa/Asir, realized by the developer during the term of this project, are used, and no further dovelopment is done by the investigator, because he has recognized that further mathematical studies and development of parallel algorithms will be of much more importance, rather than the sofrware development at this point of time. Less
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