The ALPAK System for Nonnumerical Algebra on a Digital Computer -I: Polynomials in Several Variables and Truncated Power Series with Polynomial Coefficients

Abstract

This is the first of two papers on the ALPAK system for nonnumerical algebra on a digital computer. This paper is concerned with polynomials in several variables and truncated power series with polynomial coefficients. The second paper will discuss rational functions of several variables, truncated power series with rational-function coefficients, and systems of linear equations with rational-function coefficients. The ALPAK system has been programmed within the BE-SYS-4 monitor system on the IBM 7090 computer, but the language and concepts are machine independent. The available polynomial arithmetic operations are add, subtract, multiply, divide (if divisible), substitute, differentiate, zero test, nonzero test, and equality test. The speed of the system is indicated by the rule of thumb that one man-hour equals one 7090-second. The available space in core is usually sufficient for approximately 8000 polynomial terms. Section I of this paper consists of a nontechnical description of the system and a brief glimpse into the future. Section II discusses several specific problems to which the ALPAK system has been applied. These two parts do not presuppose any knowledge of computers or computer programming. Section III describes the use and the implementation of the algebraic operations relating to polynomials in several variables and truncated power series with polynomial coefficients. The reader of Section III is assumed to be acquainted with the elements of FAP (FORTRAN Assembly Program) programming, including the use of macros, as described in a series of IBM publications, the latest of which is IBM 7090–7094 Programming Systems MAP (Macro Assembly Program) Language (Form Number C28–6311).