Study of General Formula Manipulation System

研究通用公式处理系统的结构和实现，该系统(1)执行高等数学运算，(2)统一数值、代数和图形计算，(3)配备公式数据库及其自动使用工具；(1)除了多项式和有理函数等常规数据类型外，系统还需要代数数、代数函数、集合表达式、群等多种数据类型。我们非常仔细地确定了数据类型方案，以体现清晰度，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层过程，以及将类似的作业统一到单个程序的机制。为了在不参考内部数据结构的情况下方便地实现算法，我们编写了许多抽象处理数据的程序。(2)利用OS的进程通信功能，设计了FORTRAN程序、LIP语言程序等的简单组合机制。该机制已在UNIX机上初步实现，并获得成功。此外，我们还使用数值方法和代数方法相结合地构造了几个算法。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。关于(3)，我们发现以前研究中发展的索引方法不适用于级数公式。因此，我们改进了索引，以便更好地适用于级数公式。但由于时间关系，我们未能对公式的自动使用进行研究。

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.