Computer algebra system
A computer algebra system is a software package which facilitates symbolic mathematics. Typically, these systems include
- arbitrary precision (bignum) arithmetic, allowing for instance to evaluate pi to 10,000 digits.
- symbolic manipulation engine, to simplify algebraic expressions, differentiate and integrate functions and solve equations
- graphing facility, to produce graphs of functions, typically in two and three dimensions
- linear algebra subsystem, to allow matrix computations and solution of systems of linear equations
- high level programming language, allowing the user to implement their own algorithms
The study of algorithms useful for computer algebra systems is known as computer algebra.
The run-time of numerical programs implemented in computer algebra systems is normally longer than that of equivalent programs implemented in systems such as MATLAB, octave or directly in C, because the computer algebra languages are interpreted and the bignum system may cause overhead.
Computer algebra systems began to appear in the late 1970's. The first popular system was Macsyma which is still commercially available; a copyleft version of Macsyma called GNU Maxima has also been produced. The current market leaders are Maple and Mathematica; both are commonly used by research mathematicians. MuPAD is a commercial system which provides a free version (with slightly restricted user interface) for non-commercial research and educational usage. Some computer algebra systems focus on a specific area of application; these are typically developed in academia and free. Examples include the GAP system for group theory and the Macauley and SINGULAR systems for polynomial computations and algebraic geometry.