Development of numeric-algebraic hybrid computation system

数值代数混合计算系统的开发

• 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
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-03558008/
• 关键词: approximate algebra; application of approximate algebra; formula manipulation; formula manipulation system; numeric-algebraic hybrid computation; numeric-algebraic hybrid system; 数式処理システム; 近似多項式; 数値数式融合算法; ピュイズー級数展開; 解析接続; 有理関数近似; 近似GCD; 近似Gr

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.

在本研究中，我们的目标是：1）使日本计算机代数系统GAL达到实用水平，2）为GAL提供一个数值-代数混合计算工具，3）研究近似代数算法，4）发现近似代数的各种应用。对于1）GAL已经可以作为一种研究工具，但它仍然缺乏许多通用的功能;对于2）NSL，我们进行了改造（奈良Standard-Lisp），GAL的宿主Lisp系统，使之能够连接Lisp和C语言，使我们现在能够进行数值-代数混合计算;为3），我们提出了几个重要的算法，如“代数函数的一个近似的Pu-1x级数展开算法”和“确定黎曼曲面和进行代数函数解析延拓的混合算法”;对于4），我们发现了一些有趣的应用，如“近似GCD在有理函数逼近中的应用”和“使用有理函数逼近的数据平滑”。我们得出以下结论。近似代数可以在算法层次上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.然而，近似代数算法的发展并不是看上去那么容易，需要我们付出多年的努力。特别是，我们迫切需要建立近似代数的数学理论.对于近似代数的计算系统，我们必须开发一个具有监控消号功能的浮点计算系统。

项目成果

T.Sasaki,M.Suzuki,M.Kolar and M.Sasaki: "Approximate factorization of multivariate polynomials and absolute irreducibility testing" Japan J.Indus.Appl.Math.8. 357-375 (1991)

T.Sasaki、M.Suzuki、M.Kolar 和 M.Sasaki：“多元多项式的近似因式分解和绝对不可约性检验”Japan J.Indus.Appl.Math.8。

T.Sasaki and M.Sasaki: "A unified method for multivariate polynomial factorizations" Japan J.Indus.Appl.Math.Vol.10. 21-39 (1993)

T.Sasaki 和 M.Sasaki：“多元多项式分解的统一方法”日本 J.Indus.Appl.Math.Vol.10。

Tateaki Sasaki: "A theory of approximate polynomial,I -representation and arithmetic-" Japan J.Indus.Appl.Math.11(to appear). (1994)

Tateaki Sasaki：“近似多项式的理论，I-表示和算术-”Japan J.Indus.Appl.Math.11（待出现）。

M-T.Noda,E.Miyahiro,H.Kai: "Hybrid rational function approximation and its use in the hybrid integration" Proc.IMACS Conf.on PDE7. (1993)

M-T.Noda，E.Miyahiro，H.Kai：“混合有理函数逼近及其在混合积分中的使用”Proc.IMACS Conf.on PDE7。

森継 修一,他: "行列のJordan標準形の数式処理による厳密計算法" 日本応用数理学会・論文誌. 2. (1992)

Shuichi Moritsugu 等人：“使用约旦标准矩阵数学处理的精确计算方法”日本应用数学学会杂志 2。（1992 年）