Study of General Formula Manipulation System

通用公式操作系统的研究

• 批准号: 62580029
• 负责人: SASAKI Tateaki
• 金额: $ 1.09万
• 依托单位: The Institute of Physical and Chemical Research
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1987
• 资助国家: 日本
• 起止时间: 1987 至 1989
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-62580029/
• 关键词: Formula Manipulation; General Formula Manipulation System; Abstract Data Types; Advanced Algebraic Computation; Algebraic-Numeric Computation; Formula Database; 抽象デ-タ型; 総合的数式処理; 数式処理システム; プログラムの自動生成; 数学公式データベース; 数値数式融合算法; 連立代数方程式; 浮動少数係数多項式剰余

This research is to study the structure and implementation of general formula manipulation system, which (1) performs higher mathematics, (2) unifies numeric, algebraic, and graphics computations, and (3) is equipped with formula database and its automatic utilization facility.As for (1), the system requires many data types such as algebraic number, algebraic function, set expression, group, etc., in addition to conventional data types such as polynomial and rational function. We determined the data type scheme very carefully, so as to manifest clarity and not to decrease efficiency. Furthermore, we developed a program for automatic generation of many typical top-level procedures, as well as mechanism for unifying similar jobs to a single program. In order to implement algorithms easily without referring to internal data structure, we have prepared many procedures for handling the data abstractly. By these, we succeeded in developing a kernnel of general formula manipulation system which is several times as efficient as REDUCE.As for (2), we devised a simple mechanism of combining FORTRAN programs, Lisp programs, etc. by using the process communication facility of OS. The mechanism has been implemented preliminarily and successfully on UNIX. Furthermore, we constructed several algorithms using both numeric and algebraic methods combinedly. The algorithms can solve some kinds of numerically ill-conditioned problems, and we call them "approximate algebraic algorithms".As for (3), we have found that the indexing method developed in a previous research does not work well for series formulas. So, we improved the indexing so as to work well for series formulas. However, we could not study the automatic utilization of formulas because of lack of time.

本研究旨在研究一个通用公式处理系统的结构与实现，该系统具有：（1）进行高等数学运算;（2）统一数值、代数和图形计算;（3）具有公式数据库及其自动使用功能。对于（1），系统需要代数数、代数函数、集合表达式、群等多种数据类型，除了诸如多项式和有理函数之类常规数据类型之外，还可以使用其他数据类型。我们非常仔细地确定了数据类型方案，以显示清晰性而不降低效率。此外，我们还开发了一个程序，用于自动生成许多典型的顶级程序，以及统一类似的工作到一个单一的程序的机制。为了在不涉及内部数据结构的情况下容易地实现算法，我们准备了许多处理数据抽象的过程。对于（2），我们利用操作系统的进程通信功能，设计了一种简单的FORTRAN程序与Lisp程序等程序相结合的机制，并在UNIX上初步实现。在此基础上，我们结合数值方法和代数方法构造了几种算法。这些算法可以解决某些数值病态问题，我们称之为“近似代数算法”。对于（3），我们发现以前研究中提出的索引方法对级数公式不适用。因此，我们改进了索引，以便更好地适用于系列公式。然而，由于时间不够，我们未能研究公式的自动利用。

项目成果

M.Suzuki,T.Sasaki,M.Sato,Y.Fukui: "A Hybrid Algebraic-Numeric System ANS and Its Preliminary Implementation" Lecture Notes in Computer Science. 378. 163-171 (1989)

M.Suzuki、T.Sasaki、M.Sato、Y.Fukui：“混合代数数值系统 ANS 及其初步实现”计算机科学讲义。

Masayuki Suzuki,T.Sasaki M.Sato,Y.Fukui: "A Hybrid AlgebraicーNumeris System ANS and Its Preliminary Implementation" Lecture Notes in Computer Science. 378. 163-171 (1989)

Masayuki Suzuki、T.Sasaki M.Sato、Y.Fukui：“混合代数数值系统 ANS 及其初步实现”计算机科学讲义 378. 163-171 (1989)。

Tateaki Sasaki,Mutsuko Sasaki: "Analysis of Accuracy Decreasing in Polynomial Remainder Sequence with Floatingーpoint Number Coefficients" J.Inf.Processing. 12. (1990)

Tateaki Sasaki、Mutsuko Sasaki：“浮点数系数多项式余数序列的精度下降分析”J.Inf.Processing 12。（1990）

三枝義典、増永良文、佐々木建昭: 情報処理学会・記号処理学会報告集. 89. 1-10 (1989)

Yoshinori Saegusa、Yoshifumi Masunaga、Takeaki Sasaki：日本信息处理学会/日本符号处理学会报告 89. 1-10 (1989)。

T. Sasaki and T. Takeshima: "Parallel Computation of Groebner-basis and System of Algebraic Equations (in Japanese)." Trans. Inf. Proces. Soc. Japan Vol.30, pp. 1555-1561 (1989).

T. Sasaki 和 T. Takeshima：“Groebner 基和代数方程组的并行计算（日语）”。