Computer algebra system
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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.
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 (gamma, zeta, erf, Bessel, etc.); arbitrary functions of expressions; derivatives, integrals, sums, and products of expressions; truncated series with expressions as coefficients, matrices of expressions, and so on. (This is a recursive definition.)Symbolic manipulations
The symbolic manipulations supported typically include- simplification, including automatic simplification and simplification with assumptions
- substitution of symbolic or numeric values for expressions
- change of form of expressions: expanding products and powers, rewriting as partial fractions, rewriting trigonometric functions as exponentials, etc.
- differentiation with respect to one or all variables
- symbolic constrained and unconstrained global optimization
- partial and full factorization
- solution of linear and some non-linear equations over various domains
- solution of some differential and difference equations
- taking some limits
- some indefinite and definite integration, including multidimensional integrals
- integral transforms
- expansion as truncated Taylor, Laurent and Puiseux series
- some infinite series expansion
- some series summation
- matrix operations including products, inverses, etc.
- display of mathematical expressions in two-dimensional mathematical form, often using typesetting systems similar to TeX (see also Prettyprint)
Other features
In addition, most CASs include numeric operations for:- evaluating at particular numeric values
- definite integration
- evaluating to high precision (bignum arithmetic). For instance, allowing the evaluation of 21/3 to 10,000 digits
- computations in numeric linear algebra
- plotting graphs and parametric plots of functions in two and three dimensions
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, GNU Octave, or directly in C, since they are programmed for full symbolic generality and thus cannot use machine numerical operations directly for most of their functions.
History
Computer algebra systems began to appear in the early 1970s, and evolved out of research into artificial intelligence, though the fields are now regarded as largely separate. Pioneering work was conducted by the Nobel Prize laureate Martin Veltman, who designed a program for symbolic mathematics, especially High Energy Physics, called Schoonship (Dutch for "clean ship") in 1963. The first popular systems were Reduce, Derive, and Macsyma which are still commercially available; a copyleft version of Macsyma called Maxima is actively being maintained. The current market leaders are Maple and Mathematica; both are commonly used by research mathematicians, scientists, and engineers. MuPAD and MathCad are other commercial systems. Some computer algebra systems focus on a specific area of application; these are typically developed in academia and free.Mathematics used in computer algebra systems
See also
References
- Richard J. Fateman. "Essays in algebraic simplification". Technical report MIT-LCS-TR-095, 1972. (Of historical interest in showing the direction of research in computer algebra. At the MIT LCS web site: [link])
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