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Encyclopedia > Computer algebra system

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. This article does not cite any references or sources. ... Symbolic mathematics, or symbolic math, relates to the use of computers to manipulate mathematical equations and expressions in symbolic form, as opposed to manipulating the approximations of specific numerical quantities represented by those symbols. ...

Contents

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. (This is a recursive definition.) In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ... The exponential function is one of the most important functions in mathematics. ... The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ... In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... Plot of the error function In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. ... In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: x2 for an arbitrary real or complex number α. The most common and important special case is where α is an integer n, then α is referred... In mathematics, a series is often represented as the sum of a sequence of terms. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... This article is about the concept of recursion. ...


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 symbolic, functors or numeric values for expressions
  • change of form of expressions: expanding products and powers, rewriting as partial fractions, constraint satisfaction, rewriting trigonometric functions as exponentials, etc.
  • partial and total differentiation
  • 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
  • arbitrary precision numeric operations
  • Series operations such as expansion, summation and products
  • 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)

(In the above, the word some indicates that the operation cannot always be performed.) For functors in computer science, see the function object article. ... In artificial intelligence and operations research, constraint satisfaction is the process finding a solution to a set of constraints. ... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ... In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant. ... In mathematics (more precisely in differential calculus), the term total derivative has a number of closely related meanings. ... ... In mathematics, an equation or system of equations is said to have a closed-form solution just in case a solution can be expressed analytically in terms of a bounded number of well_known operations. ... A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ... In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ... In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ... In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ... This article is about the concept of integrals in calculus. ... Arbitrary precision mathematical libraries allow computer programs to perform calculations and then specify how many digits will be used for the result. ... In mathematics, a series is a sum of a sequence of terms. ... It has been suggested that this article or section be merged with invertible matrix. ... TeX (IPA: as in Greek, often in English; written with a lowercase e in imitation of the logo) is a typesetting system created by Donald Knuth. ... To prettyprint (or pretty-print) is to present an object to a human reader, so that it is easier to perceive the objects structure, or, less commonly, to simply make it more attractive. ...


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. A bignum package in a computer or program allows internal representation of very large integers, rational numbers, decimal numbers, or floating-point numbers (limitted only by available memory), and provides a set of arithmetic operations on such numbers. ... Add-ons are optional computer hardware or software modules that supplement or enhance the original unit they are adding on to. ... Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Computational physics is the study and implementation of numerical algorithms in order to solve problems in physics for which a quantitative theory already exists. ... Plotting is the process of depicting mathematical functions or data visually. ... API and Api redirect here. ... String manipulation is the process in computer programming languages for handling, matching, parsing, searching or formatting of character strings. ... String searching algorithms, sometimes called string matching algorithms, are an important class of string algorithms that try to find a place where one or several strings (also called patterns) are found within a larger string or text. ... String searching algorithms are an important class of string algorithms that try to find a place where one or several strings (also called patterns) are found within a larger string or text. ... This is an incomplete list of software that is designed for the explicit purpose of performing statistical analyses. ... Automated theorem proving (currently the most important subfield of automated reasoning) is the proving of mathematical theorems by a computer program. ... Proof checking is the process of using software for checking proofs for correctness. ... For the journal by ACM SIGGRAPH, see Computer Graphics (Publication). ... The seawater creature in The Abyss marked CGIs acceptance in the visual effects industry. ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... UPIICSA IPN - Binary image Image processing is any form of information processing for which the input is an image, such as photographs or frames of video; the output is not necessarily an image, but can be for instance a set of features of the image. ... The term synthesiser is also used to mean frequency synthesiser, an electronic system found in communications. ... A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ... Bold textConstraint logic programming is a form of constraint programming, in which logic programming is extended to include concepts from constraint satisfaction. ... b fourth-generation programming language(1970s-1990) (abbreviated 4GL) is a programming language or programming environment designed with a specific purpose in mind, such as the development of commercial business software. ... A fifth-generation programming language (abbreviated 5GL) is a programming language based around solving problems using constraints given to the program, rather than using an algorithm written by a programmer. ...


The study of algorithms 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. In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ...


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. Not to be confused with Matlab Upazila in Chandpur District, Bangladesh. ... Octave is a free computer program for performing numerical computations which is mostly compatible with MATLAB. It is part of the GNU project. ... C is a general-purpose, block structured, procedural, imperative computer programming language developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system. ...


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. The 1970s decade refers to the years from 1970 to 1979, also called The Seventies. ... Bold text[[Link title]] “AI” redirects here. ... Martinus J.G. Veltman (Tini for short) (born June 27, 1931, Waalwijk) is a 1999 Nobel Prize in Physics laureate for elucidating the quantum structure of electroweak interactions in physics, work done at Utrecht University, The Netherlands. ... REDUCE is a general-purpose computer algebra system geared towards applications in physics. ... Derive is a computer algebra system, developed as a successor to muMATH by the Soft Warehouse in Honolulu, Hawaii, now owned by Texas Instruments. ... MACSYMA Reference Manual, MIT, 1977 Macsyma is a computer algebra system that was originally developed from 1967 to 1982 at MIT as part of Project MAC and later marketed commercially. ... The reversed c in a full circle is the copyleft symbol. ... For other uses, see Maxima (disambiguation). ... Maple 9. ... For other uses, see Mathematica (disambiguation). ... MuPAD is a commercial computer algebra system developed by the MuPAD research group at the University of Paderborn in Paderborn, Germany, under the direction of Professor Benno Fuchssteiner, in cooperation with SciFace Software GmbH. Its syntax is modelled loosely on the Maple computer algebra system. ... Mathcad (originally written MathCAD) is desktop software for performing and documenting engineering and scientific calculations. ...


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 Hewlett-Packard Company (NYSE: HPQ), commonly known as HP, is a very large, global company headquartered in Palo Alto, California, United States. ... The HP-28C and HP-28S were two graphing calculators produced by Hewlett-Packard from 1987 to 1992. ...


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. Texas Instruments (NYSE: TXN), better known in the electronics industry (and popularly) as TI, is an American company based in Dallas, Texas, USA, renowned for developing and commercializing semiconductor and computer technology. ... The Texas Instruments TI-92 calculator, originally released in 1995, was a large calculator with a QWERTY keyboard. ... Derive is a Computer algebra system, developed as a successor to MuMATH by the Soft Warehouse in Honolulu, Hawaii, now owned by Texas Instruments. ... The TI-89 and the TI-89 Titanium are graphing calculators developed by Texas Instruments. ...


Mathematics used in computer algebra systems

Symbolic integration is the application of computer software to solving problems in mathematics of find the integral of an expression, but finding an expression rather than a value. ... In computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating subset of an ideal I in a polynomial ring R. One can view it as a multivariate, non-linear generalization of: the Euclidean algorithm for computation of univariate greatest common... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ... ...

CAS software

XCAS or on-call close air support (CAS) is a technique of air warfare which first achieved recognition in Operation Enduring Freedom, the US-led invasion of Afghanistan in 2001. ...

See also

The following tables provide a comparison of computer algebra systems (CAS). ... Scientific computation is a term often confused with scientific computing. ... A statistical package is a kind of large computer program that is specialised for statistical analysis. ... Symbolic computation, computer algebra, algebraic computation, or, less commonly, symbolic manipulation, symbolic processing, symbolic mathematics, or symbolic algebra, relates to the use of machines, such as computers, to manipulate mathematical equations and expressions in symbolic form, as opposed to manipulating the approximations of specific numerical quantities represented by those symbols. ... Symbolic computation relates to the use of computers to manipulate mathematical equations and expressions in symbolic form, as opposed to manipulating the approximations of specific numerical quantities represented by those symbols. ... Automated theorem proving (ATP) or automated deduction, currently the most well-developed subfield of automated reasoning (AR), is the proving of mathematical theorems by a computer program. ... Bold text[[Link title]] “AI” redirects here. ... Constraint logic programming is a form of constraint programming, in which logic programming is extended to include concepts from constraint satisfaction. ...

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: [1])

  Results from FactBites:
 
Computer algebra system - Wikipedia, the free encyclopedia (453 words)
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.
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.
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