Computer Algebra

Concepts and Techniques

Edmund A. Lamagna

• Publisher: CRC Press
• Year: 2019
• ISBN:
  9781351605823
• Edition: 1
• Topic: Mathématiques
• Subtopic: Mathématiques générales

The goal of Computer Algebra: Concepts and Techniques is to demystify computer algebra systems for a wide audience including students, faculty, and professionals in scientific fields such as computer science, mathematics, engineering, and physics. Unlike previous books, the only prerequisites are knowledge of first year calculus and a little programming experience — a background that can be assumed of the intended audience. The book is written in a lean and lively style, with numerous examples to illustrate the issues and techniques discussed. It presents the principal algorithms and data structures, while also discussing the inherent and practical limitations of these systems

Chapter 1

Introduction

The last thing one knows when writing a book is what to put first.

Blaise Pascal
(1623–1662) Pensées, 1660

How thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which, at the same time, assist in understanding earlier theories and in casting aside some more complicated developments.

David Hilbert
(1862–1943) International Congress of Mathematicians, Paris, 1900

Computer algebra systems (CAS) provide impressive mathematical capabilities that have revolutionized the way mathematics is done and taught. They can perform without error gargantuan calculations that would be infeasible to attempt by hand. Accordingly, CAS have become commonplace computational and modeling tools in the scientific and engineering workplace. They have also been incorporated into the mathematics curricula at many colleges and universities starting with calculus and moving to more advanced topics such as linear algebra and differential equations. Yet despite their widespread application and importance, few users are familiar with the inner workings of CAS. The goal of this book is to demystify CAS for a wide audience that includes students, faculty, and professionals in scientific fields such as computer science, mathematics, engineering, and physics.

1.1 What is Computer Algebra?

Computer algebra is a branch of scientific computation that lies at the intersection of computer science and mathematics. It is readily distinguished from numerical analysis, the other principal branch of scientific computation, in that it involves the manipulation of formulas rather than only numbers. An article in the Encyclopedia of Computer Science (2000) by Gaston H. Gonnet and Dominik W. Gruntz defines computer algebra as “computation with variables and constants according to the rules of algebra, analysis, and other branches of mathematics.” The authors go on to describe computer algebra as “formula manipulation involving symbols, unknowns, and formal operations rather than with conventional computer data of numbers and character strings.”

Computer algebra systems can be classified as either special purpose or general purpose. Special purpose systems are designed to solve problems in one specific area of science or mathematics, such as celestial mechanics or group theory. General purpose systems are, of course, intended for use in most scientific and mathematical fields. General systems are typically used interactively and provide the following facilities:

• symbolic computation: expansion and factorization of polynomials, differentiation, indefinite and exact definite integration, exact solution of equations and systems of equations, and linear algebra

• numerical computation: arbitrary precision numerical computation including definite integration, numerical solution of equations, and evaluation of elementary and special functions

• graphics: plotting in two and three dimensions

• programming: an embedded programming language and data structures allowing users to extend the system

The focus of this book is on the symbolic capabilities of computer algebra systems. We present several examples that illustrate the types of calculations these systems can perform and which distinguish them from other mathematical software. First, CAS perform exact arithmetic and can make transformations such as

$\begin{array}{cc}\frac{1}{2}+\frac{1}{3}\to \frac{5}{6},& \frac{1}{{\left(\sqrt{2}-1\right)}^2}\to 3+2\sqrt{2}\end{array}.$

Most calculators would give only the numerical approximations 0.833333 … and 5.82843 … to these quantities. Moreover, CAS perform calculations with essentially unlimited precision and can therefore produce results like

$\begin{array}{l}100!\to 9332621544394415268169923885626670049071596826438162146\\ 8592963895217599993229915608941463976156518286253697\\ 920827223758251185210916864000000000000000000000000.\end{array}$

Another important capability of computer algebra systems is their ability to manipulate polynomials and rational functions (ratios of polynomials). CAS can expand polynomials by multiplying out products,

$(x^2-4x+4)(x^2+4x+4)\to x^4-8x^2+16,$

factor polynomials,

$(x^2-4x+4)(x^2+4x+4)\to {(x-2)}^2{(x+2)}^2,$

and simplify polynomial ratios by removing the greatest common divisor,

$\frac{x^4-1}{x^8-1}\to \frac{1}{x^4+1}.$

Computer algebra systems can also perform exact computations with elementary transcendental functions (exponentials, logarithms, trigonometric and inverse trigonometric functions). For example,

$\begin{array}{cc}\sin (\pi /3)\to \frac{\sqrt{3}}{2},& \arccos (\sqrt{2}/2)\to \frac{\pi }{4}\end{array},$

rather than 0.866025 … and 0.785398 …. CAS can apply multiangle trigonometric identities in either direction,

$\sin (3x)\leftrightarrow 4\sin x{(\cos x)}^2-\sin x,$

as well as expand and factor polynomials of functions,

$\begin{array}{c}(e^{3x}-1)(e^{2x}-1)\to e^{5x}-e^{3x}-e^{2x}+1,\\ (e^{3x}-1)(e^{2x}-1)\to {(e^x-1)}^2(e^x-1)(e^{2x}+e^x+1).\end{array}$

Another feature of computer algebra systems is their ability to perform the basic operations of calculus (differentiation, indefinite and exact definite integration, calculation of limits, series expansion, and summation) and linear algebra. For instance, CAS can transform

$\frac{d}{dx}{x}^2\cos x\to 2x\cos x-x^2\sin x$

and

${\displaystyle \int x{e}^{x}dx\to x{e}^x-e^x,}$

and determine that ${\displaystyle \int e^{x^2}dx}$ cannot be expressed with only elementary functions.

The term comput...