Formal calculation

Formal computation is now widely recognized in the scientific community in general and among engineers in particular. Indeed, these days it is easy to install and use a formal computation system on a simple, low-cost personal computer (PC, Macintosh). Once you have acquired such a system, you need to learn how to use it. At first, it is very easy to perform simple calculations, similar to those of a “formal calculator,” but to go further, some knowledge of the system and its limitations is necessary. Otherwise, the user quickly becomes discouraged and gives up. Therefore, training time is essential for using a formal computation system.

This raises the question: “Is symbolic computation useful to me?”; in other words, “Is it worth my while to spend time learning how to use such a system?” The purpose of this article is to answer that question. To do so, we will review the main areas of mathematics in which symbolic computation can solve problems. These fields are those in which engineers typically work: calculations involving rational numbers and fractions, differentiation, simplifying formulas, and plotting curves—which form the basis of any formal computation system—as well as integral and matrix calculations, and solving nonlinear equations and systems of differential equations commonly used by engineers. And finally, we must discuss numerical computation. The latter is generally the culmination of an engineer’s work, and formal computation should not be pitted against numerical computation. We will indeed demonstrate cases where formal computation can prove very useful in this field. For each section, we will show what formal computation can do, how it does it, and what its limitations are.

This document includes numerous examples to illustrate how symbolic computation works within a system. We have chosen the Maple symbolic computation system for this purpose. The reason is that this system is widely used (like Mathematica), has a sufficiently rich and open library (the source code for most functions is available), and is easily extensible.

The way a symbolic computation system like Maple works is simple: the user enters a command, ending with a semicolon “;” in a very natural syntax, and Maple displays the result in a high-resolution format that resembles mathematical typesetting. If you replace the semicolon with a colon “:”, the result is not displayed. Maple uses the concept of packages, meaning that a large number of commands are organized into groups based on their functionality. In this case, the command is written as [], such as LinearAlgebra [Determinant].

The purpose of this article is not to describe the Maple computer algebra system. We will not explain...