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TYPES OF EXPRESSIONS The expressions manipulated by the CAS typically include Polynomial s in multiple variables; standard functions of expressions ( Sine , Exponential , etc.); various special functions ( Γ , ζ , Erf , Bessel Function s, 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. (This is a Recursive definition.) SYMBOLIC MANIPULATIONS The symbolic manipulations supported typically include
(In the above, the word ''some'' indicates that the operation cannot always be performed.) OTHER FEATURES Many CASs have additional features:
Many also include a high level Programming Language , allowing users to implement their own algorithms. The programming language may be similar to a conventional imperative or functional programming language or a higher-level Constraint Logic , Fourth-generation or Fifth-generation Programming Language . The study of Algorithm s useful for computer algebra systems is known as computer algebra, '''symbolic computation''', '''algebraic computation''', or, less commonly, '''symbolic manipulation''', '''symbolic processing''', '''symbolic mathematics''', or '''symbolic algebra'''. The run-time of some ''numerical'' programs implemented in some computer algebra systems is 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 or optimize into direct machine numerical operations for most of their functions. On the other hand, some systems can compile user-written numerical programs into efficient running code if the user supplies suitable declarations. 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. 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) feature a reasonably capable and relatively inexpensive handheld computer algebra system, featuring derivatives and integrals with respect to 1 variable, limits, and some differential equations. MATHEMATICS USED IN COMPUTER ALGEBRA SYSTEMS SEE ALSO
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