General-purpose computer algebra systems (CASs) such as AXIOM, DERIVE, MACSYMA, MAPLE, Mathematica, MuPAD, and REDUCE are large computer programs for solving symbolic mathematical problems. Each of these systems has its strong points and limitations, and its own following among members of the technical community.
The goal of this book is to describe the possibilities and limitations of this software; evaluate, compare, and contrast the capabilities of various computer algebra systems; give an overview of the theoretical foundations and design issues in a number of areas; provide a historical perspective; and to suggest how this software might evolve. Mostly, the book helps users of this software cut through all the hype and inflated claims so that they can make an informed decision about the suitability of the software to solve a mathematical problem.
This book includes articles by specialists in the development and use of CAS software, and is the most balanced and realistic discussion of these systems that I have seen. Topics include commercial software; a comparison of system performance on collections of problems; CAS equation solvers; technical issues in expression evaluation, the denesting of expressions with square roots, limit computation, the solution of ordinary differential equations, and code generation for FORTRAN and C; some application areas, namely real and complex analysis, computation of Chebyshev polynomials, and integrability tests for nonlinear evolution equations; the symbolic aspects of Babbage’s analytical engine; and a review of CAS applications in mathematical education.
Since computer algebra systems are developing and changing rapidly, it is risky to make statements about system performance that might change with the next software release. Thus, the importance of this book is not so much in the actual comparisons of systems as in the questions that are raised about the software. To quote the editor, “It is the mark of a good review, though, if it makes people think.” Although I have used this software for many years, the book raises many issues I had not considered. It is an important addition to the literature on computer algebra, and a useful reference for mathematicians, computer scientists, engineers, scientists, educators, and others who use this software.
