Development of numeric-algebraic hybrid computation system

点击翻译按钮获取中文摘要

In this research, we aimed at 1) bringing Japanese computer algebra system GAL up to practically usable level, 2) providing a numeric-algebraic hybrid computation facility to GAL, 3) studyng approximate algebraic algorithms, and 4) finding various applications of approximate algebra. As 1), GAL is now usable as a resesarch tool but it still lacks many facilities for general use ; as for 2), we reformed NSL (Nara Standard-Lisp), the host Lisp system for GAL, so that it can link Lisp and C, enabling us to perform numeric-algebraic hybrid computation now ; was for 3), we developed several important algorithms such as "an approximate Puiseux series expansion algorithm of algebraic functions" and "a hybrid algorithm for determining Riemann surface and performing analytic continuation of algebraic functions" ; as for 4), several interesting applications were found such as "application of approximate GCD to rational function approximation" and "sumoothing of data using the rational function approximation".Summarizing our research, we have the following conclusions.1. Approximate algebra can fuse numeric and algebraic conputation in algorithm level, and it has a very good possibility of innovating scientific computation.2. However, development of approximate algebraic algorithm is not so easy at it looks, requesting us to make many years of efforts. In particular, we must construct mathematical theory of approximate algebra urgently.3. As for computation system of approximate algebra, we must develop a floating-point computation system with facility of monitoring cancellation of numbers.