Study of General Formula Manipulation System

本研究研究通用公式操作系统的结构和实现，该系统(1)执行高等数学，(2)统一数值、代数和图形计算，(3)配备公式数据库及其自动利用设施。对于(1)，系统除了需要多项式、有理函数等常规数据类型外，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，以便表现出清晰而不降低效率。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过程，以及将类似的工作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轻松实现算法，我们准备了许多抽象处理数据的过程。通过这些，我们成功地开发了一个通用公式处理系统的内核，其效率是REDUCE的几倍。对于(2)，我们设计了一种简单的机制，利用OS的进程通信功能将FORTRAN程序、Lisp程序等结合起来。该机制已在UNIX上初步实现并取得了成功。此外，我们还结合数值和代数方法构造了几种算法。该算法可以解决一些数值病态问题，我们称其为“近似代数算法”。对于(3)，我们发现前人研究的索引方法对于序列公式的效果并不好。因此，我们对索引进行了改进，使其能很好地适用于级数公式。但是由于时间的关系，我们无法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.