Development of numeric-algebraic hybrid computation system

数值代数混合计算系统的开发

基本信息

批准号：
03558008
负责人：
SASAKI Tateaki
金额：
$ 4.16万
依托单位：
University of Tsukuba
依托单位国家：
日本
项目类别：
Grant-in-Aid for Developmental Scientific Research (B)
财政年份：
1991
资助国家：
日本
起止时间：
1991 至 1993
项目状态：
已结题

项目摘要

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.

在这项研究中，我们的目标是 1) 将日本计算机代数系统 GAL 提升到实用水平，2) 为 GAL 提供数值代数混合计算工具，3) 研究近似代数算法，以及 4) 寻找近似代数的各种应用。如1），GAL现在可以用作研究工具，但仍然缺乏许多通用设施； 2）我们对GAL的宿主Lisp系统NSL（Nara Standard-Lisp）进行了改造，使其能够链接Lisp和C，使我们现在能够进行数值代数混合计算；针对3），我们开发了“一种代数函数的近似Puiseux级数展开算法”和“一种确定黎曼曲面并执行代数函数解析延拓的混合算法”等几个重要算法；对于4），发现了一些有趣的应用，例如“近似GCD在有理函数逼近中的应用”和“使用有理函数逼近对数据进行求和”。总结我们的研究，我们得到以下结论： 1．近似代数可以在算法层面融合数值计算和代数计算，具有很好的创新科学计算的可能性。 2．然而，近似代数算法的开发并不是表面上那么容易，需要我们付出多年的努力。尤其是近似代数的数学理论亟待构建。 3．对于近似代数的计算系统，我们必须开发一个具有监测数字消去功能的浮点计算系统。