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Computer algebra system

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A computer algebra system (CAS) is a software for computer algebra, that is a program, typically interactive, that allows to manipulate and compute with formulas, mathematical expressions and other mathematical objects, and compute with them. The main functionality of a CAS is the ability of doing all operations of elementary algebra, but the main computer algebra systems have much more possibilities, that include most methods of computation that have been developed in algebra, but also in calculus and many areas of mathematics.

CAS are typically complex softwares (therefore the word "system") that include

  • an user interface allowing to enter and editing formulas and to display formulas in two dimension
  • a method to represent complex expressions containing mathematical variables that do not have any numeric value
  • a memory manager
  • basic functions to evaluate, simplify and transform these expressions
  • a large library of functions for the various operations that the user may want to do.
  • a programming language allowing the user to combine several operations for implementing his own algorithms.

These different aspects may not be developed independently: for example, the simplification of fractions requires an efficient program for multivariate polynomial greatest common divisor, and a simplifier for trigonometric functions may need a Gröbner basis computation.

Symbolic manipulations

The symbolic manipulations supported typically include:

In the above, the word some indicates that the operation cannot always be performed.

Additional capabilities

Many also include:

Some include:

Some computer algebra systems focus on a specific area of application; these are typically developed in academia and are free. They can be inefficient for numeric operations compared to numeric systems.

Types of expressions

The expressions manipulated by the CAS typically include polynomials in multiple variables; standard functions of expressions (sine, exponential, etc.); various special functions (Γ, ζ, erf, Bessel functions, etc.); arbitrary functions of expressions; optimization; derivatives, integrals, simplifications, sums, and products of expressions; truncated series with expressions as coefficients, matrices of expressions, and so on. Numeric domains supported typically include real, complex, interval, rational, and algebraic.

History

Computer algebra systems began to appear in the 1960s, and evolved out of two quite different sources - the requirements of theoretical physicists and research into artificial intelligence.

A prime example for the first development was the pioneering work conducted by the later Nobel Prize laureate in physics Martin Veltman, who designed a program for symbolic mathematics, especially High Energy Physics, called Schoonschip (Dutch for "clean ship") in 1963.

Using LISP as the programming basis, Carl Engelman created MATHLAB in 1964 at MITRE within an artificial intelligence research environment. Later MATHLAB was made available to users on PDP-6 and PDP-10 Systems running TOPS-10 or TENEX in universities. Today it can still be used on SIMH-Emulations of the PDP-10. MATHLAB ("mathematical laboratory") should not be confused with MATLAB ("matrix laboratory") which is a system for numerical computation built 15 years later at the University of New Mexico, accidentally named rather similarly.

The first popular computer algebra systems were muMATH, Reduce, Derive (based on muMATH), and Macsyma; a popular copyleft version of Macsyma called Maxima is actively being maintained. As of today, the most popular commercial systems are Mathematica[1] and Maple, which are commonly used by research mathematicians, scientists, and engineers. Freely available alternatives include Sage (which can act as a front-end to several other free and nonfree CAS).

In 1987 Hewlett-Packard introduced the first hand held calculator CAS with the HP-28 series, and it was possible, for the first time in a calculator, to arrange algebraic expressions, differentiation, limited symbolic integration, Taylor series construction and a solver for algebraic equations.

The Texas Instruments company in 1995 released the TI-92 calculator with an advanced CAS based on the software Derive. This, along with its successors (including the TI-89 series and the newer TI-Nspire CAS released in 2007) featured a reasonably capable and inexpensive hand-held computer algebra system.

CAS-equipped calculators are not permitted on the ACT, the PLAN, and in some classrooms because they may affect the integrity of the test/class,[2] though it may be permitted on all of College Board's calculator-permitted tests, including the SAT, some SAT Subject Tests and the AP Calculus, Chemistry, Physics, and Statistics exams.

Mathematics used in computer algebra systems

See also

References

  1. ^ Interview with Gaston Gonnet, co-creator of Maple, SIAM History of Numerical Analysis and Computing, March 16, 2005
  2. ^ ACT's CAAP Tests: Use of Calculators on the CAAP Mathematics Test


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