Computer algebra system
From Wikipedia, the free encyclopedia
A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form.
Contents 
[edit] Symbolic manipulations
The symbolic manipulations supported typically include:
 simplification to the smallest possible expression or some standard form, including automatic simplification with assumptions and simplification with constraints
 substitution of symbols, functors or numeric values for expressions
 change of form of expressions: expanding products and powers, partial and full factorization, rewriting as partial fractions, constraint satisfaction, rewriting trigonometric functions as exponentials, etc.
 partial and total differentiation
 symbolic constrained and unconstrained global optimization
 solution of linear and some nonlinear equations over various domains
 solution of some differential and difference equations
 taking some limits
 some indefinite and definite integration, including multidimensional integrals
 integral transforms
 series operations such as expansion, summation and products
 matrix operations including products, inverses, etc.
 addons for use in applied mathematics such as physics packages for physical computation
 statistical computation
 theorem proving and verification
In the above, the word some indicates that the operation cannot always be performed.
[edit] Additional capabilities
Many also include:
 a programming language, allowing users to implement their own algorithms
 arbitraryprecision numeric operations
 display of mathematical expressions in twodimensional mathematical form, often using typesetting systems similar to TeX (see also Prettyprint)
 plotting graphs and parametric plots of functions in two and three dimensions, and animating them
 drawing charts and diagrams
 APIs for linking it on an external program such as a database, or using in a programming language to use the computer algebra system
 string manipulation such as matching and searching
Some include:
 graphic production and editing such as CGI and signal processing as image processing
 sound synthesis
Some computer algebra systems focus on a specific area of application; these are typically developed in academia and are free. They can be relatively inefficient for numeric operations compared to numeric systems.
[edit] 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.
[edit] 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.
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 programming basis, Carl Engelman created MATHLAB in 1964 at MITRE within an artificial intelligence research environment. Later MATHLAB was made available to users on PDP6 and PDP10 Systems running TOPS10 or TENEX in universities. Today it can still be used on SIMHEmulations of the PDP10. 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 like Sage (a frontend to several free CAS) and Xcas are also gaining popularity.
In 1987 HewlettPackard introduced the first hand held calculator CAS with the HP28 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 TI92 calculator with an advanced CAS based on the software Derive. This, along with its successors (including the TI89 series and the newer TINspire CAS released in 2007) featured a reasonably capable and relatively inexpensive handheld computer algebra system.
[edit] Mathematics used in computer algebra systems
 Symbolic integration
 Gröbner basis
 Greatest common divisor
 Polynomial factorization
 Risch algorithm
 Cylindrical algebraic decomposition
 CantorZassenhaus algorithm
 Padé approximant
 SchwartzZippel lemma and testing polynomial identities
 Chinese remainder theorem
 Gaussian elimination
 Diophantine equations
[edit] See also
 Comparison of computer algebra systems
 Scientific computation
 Statistical package
 Algebraic algorithms
 Symbolic computation
 Automated theorem proving
 Artificial intelligence
 Constraintlogic programming
 App4Math, a CAS developed for the TI83 series of graphing calculators
[edit] References
 ^ Interview with Gaston Gonnet, cocreator of Maple, SIAM History of Numerical Analysis and Computing, March 16, 2005
[edit] External links
 Definition and workings of a computer algebra system
 Curriculum and Assessment in an Age of Computer Algebra Systems  From the Education Resources Information Center Clearinghouse for Science, Mathematics, and Environmental Education, Columbus, Ohio.
 Richard J. Fateman. "Essays in algebraic simplification". Technical report MITLCSTR095, 1972. (Of historical interest in showing the direction of research in computer algebra. At the MIT LCS web site: [1])
