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The ALPAK System for Nonnumerical Algebra on a Digital Computer - III: Systems of Linear Equations and a Class of Side Relations

01 July 1964

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This is the third and last in a series of papers describing the ALPAK system, a programming system for performing routine manipulations of algebraic expressions on a digital computer. The system can perform the operations of addition, subtraction, multiplication, division, sub1547 1548 T H E B E L L SYSTEM TECHNICAL J O U R N A L , J U L Y 1964 stitution, and differentiation. The first paper 1 is concerned with polynomials in several variables and truncated power series with polynomial coefficients. The second paper 2 is concerned with rational functions in several variables and truncated power series with rational-function coefficients. The present paper describes the ALPAK facilities for manipulating and solving by Gaussian elimination systems of equations linear in certain variables with coefficients which are rational functions of other variables. The facilities for handling a certain class of side relations are also described. The ALPAK system lias been programmed within the BE-SYS-4 monitor system on the IBM 7090 computer, but the language and concepts are machine independent. Several practical applications are described in Ref. 1. This paper is divided into six sections, of which the first two do not presuppose any knowledge of computers or computer programming and the last four assume that the reader is familiar with the basic concepts of computer programming and Refs. 1 and 2. Section I is a general description of ALPAK and deals with basic concepts. Section II describes the different forms in which a linear system can occur, including especially the canonical form of a linear system.