The journal Computing has established a series of supplement volumes the fourth
of which appears this year. Its purpose is to provide a coherent presentation of a
new topic in a single volume. The previous subjects were Computer Arithmetic
1977, Fundamentals of Numerical Computation 1980, and Parallel Processes and
Related Automata 1981; the topic of this 1982 Supplementum to Computing is
Computer Algebra. This subject, which emerged in the early nineteen sixties, has
also been referred to as "symbolic and algebraic computation" or "formula
manipulation".
Algebraic algorithms have been receiving increasing interest as a result of the
recognition of the central role of algorithms in computer science. They can be easily
specified in a formal and rigorous way and provide solutions to problems known
and studied for a long time. Whereas traditional algebra is concerned with
constructive methods, computer algebra is furthermore interested in efficiency, in
implementation, and in hardware and software aspects of the algorithms. It
develops that in deciding effectiveness and determining efficiency of algebraic
methods many other tools —recursion theory, logic, analysis and combinatorics,
for example - are necessary. In the beginning of the use of computers for symbolic
algebra it soon became apparent that the straightforward textbook methods were
often very inefficient. Instead of turning to numerical approximation methods,
computer algebra studies systematically the sources of the inefficiency and searches
for alternative algebraic methods to improve or even replace the algorithms.