Computer Algebra Systems

ID: 88d59a87-da3b-4121-aa3c-541211f304a7
CREATED: <2025-05-08 Thu 18:05>

[2025-05-08 Thu 18:06] – wiki [2025-05-08 Thu 18:07] – Maxima – Computer Algebra Systems

• Symbolic Mathematics Using Maxima
• Simplifier: https://maxima.sourceforge.io/misc/Fateman-Salz_Simplifier_Paper.pdf
• Internal forms of data in Macsyma

There are two external forms of data in Macsyma visible to the user, and three (sometimes more) internal forms. The external form for input is a string of ASCII characters as typed in by the user or read from a file. (For “example, the string "y+3/x+y”). The output display of characters on a two-dimensional "grid" is the other external form, for example,

3 2y + —. x

Between the external forms there are several internal forms. In actuality each is a Lisp "S-expression," a tree-like nested structure of list cells, pointers, and atomic names. The printed forms of these, expressed in the Lisp language, will usually look like parenthesized prefix strings. The different internal forms are:

(1) The output from the parser, which is handed to the command interpreter.

(2) The simplified form which most commands obtain by evaluating and simplifying their arguments.

(3) The formatted form, which the display program uses to translate from the simplified form to a more "user oriented" expression. In the Berkeley VAX/UNIX version of the simplifier/display, a fourth form, namely typesetter code, is also used. (There can be other forms, e.g. TeX form, rational form..)

Some commands, as have been previously noted, impose different notions of simplification on expressions. We will not deal with these in any detail, although it is part of the overall philosophy of Macsyma that alternative forms are allowed to co-exist, or can be converted to one another, in order to provide the most storage-efficient data structure and/or provide the fastest type of computation.

By and large the data represents algebraic expressions, but sometimes represents programs, program fragments, messages, names of variables, files, or other information.

The simplification program, SIMPLIFYA, is usually invoked by manipulatory functions such as the integration package. The input is an algebraic expression written in a type of parenthesized prefix notation. The input may have been partly simplified in the past, or may be "raw" from the parser. An expression is either an integer or floating-point number, an atomic "indeterminate," or a Lisp Sexpression of the form (OPERATOR . ARGLIST). The ARGLIST is a list in the Lisp meaning, which contains the appropriate number of arguments for the associated OPERATOR. Each argument is itself (with a few exceptions having to do with the OPERATORs MRAT and $POIS) an algebraic expression. Many operators have a fixed number of arguments. The exceptions include "n-ary" operators like PLUS, TIMES and DERIVATIVE.

There is an initial collection of OPERATORs known to the simplifier; these may be augmented by using the "TELLSIMP" commands in Macsyma discussed in section B.

The symbols in the OPERATORs are a mixture of historical conventions, and several have two or more variants which are intended to signify the difference between a "verb" (e.g. Integrate this expression) and a "noun" (e.g. Consider the integral …). The first part of each OPERATOR (the Lisp CAR) is indicative of the algebraic meaning. The rest of the OPERATOR consists of "flags" describing the arguments, or modifying the meaning of the OPERATOR. We make these notions more definite by examples in the next sections.