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A computer algebra system (CAS) or symbolic algebra system (SAS) is any
mathematical software Mathematical software is software used to model, analyze or calculate numeric, symbolic or geometric data. Evolution of mathematical software Numerical analysis and symbolic computation had been in most important place of the subject, but other ...
with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History O ...
s and
scientist A scientist is a person who conducts scientific research to advance knowledge in an area of the natural sciences. In classical antiquity, there was no real ancient analog of a modern scientist. Instead, philosophers engaged in the philosoph ...
s. The development of the computer algebra systems in the second half of the 20th century is part of the discipline of "
computer algebra In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressio ...
" or "
symbolic computation In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expression ...
", which has spurred work in
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s over
mathematical objects A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical ...
such as
polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
s. Computer algebra systems may be divided into two classes: specialized and general-purpose. The specialized ones are devoted to a specific part of mathematics, such as
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
,
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
, or teaching of
elementary mathematics Elementary mathematics consists of mathematics topics frequently taught at the primary or secondary school levels. In the Canadian curriculum, there are six basic strands in Elementary Mathematics: Number, Algebra, Data, Spatial Sense, Fina ...
. General-purpose computer algebra systems aim to be useful to a user working in any scientific field that requires manipulation of mathematical expressions. To be useful, a general-purpose computer algebra system must include various features such as: *a
user interface In the industrial design field of human–computer interaction, a user interface (UI) is the space where interactions between humans and machines occur. The goal of this interaction is to allow effective operation and control of the machine fr ...
allowing a user to enter and display mathematical formulas, typically from a keyboard, menu selections, mouse or stylus. *a
programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming ...
and an interpreter (the result of a computation commonly has an unpredictable form and an unpredictable size; therefore user intervention is frequently needed), *a simplifier, which is a
rewrite system In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines, or redu ...
for simplifying mathematics formulas, *a memory manager, including a
garbage collector A waste collector, also known as a garbageman, garbage collector, trashman (in the US), binman or (rarely) dustman (in the UK), is a person employed by a public or private enterprise to collect and dispose of municipal solid waste (refuse) and ...
, needed by the huge size of the intermediate data, which may appear during a computation, *an
arbitrary-precision arithmetic In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are li ...
, needed by the huge size of the integers that may occur, *a large library of mathematical
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s and
special functions Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by ...
. The library must not only provide for the needs of the users, but also the needs of the simplifier. For example, the computation of
polynomial greatest common divisor In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common d ...
s is systematically used for the simplification of expressions involving fractions. This large amount of required computer capabilities explains the small number of general-purpose computer algebra systems. Significant systems include
Axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
, Maxima,
Magma Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natural s ...
,
Maple ''Acer'' () is a genus of trees and shrubs commonly known as maples. The genus is placed in the family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated since http ...
,
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...
, and
SageMath SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, numerical analysis, n ...
.


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 Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech re ...
. A prime example for the first development was the pioneering work conducted by the later Nobel Prize laureate in physics
Martinus Veltman Martinus Justinus Godefriedus "Tini" Veltman (; 27 June 1931 – 4 January 2021) was a Dutch theoretical physicist. He shared the 1999 Nobel Prize in physics with his former PhD student Gerardus 't Hooft for their work on particle theory. Biog ...
, who designed a program for symbolic mathematics, especially high-energy physics, called
Schoonschip Schoonschip was one of the first computer algebra systems, developed in 1963 by Martinus J. G. Veltman, for use in particle physics. "Schoonschip" refers to the Dutch expression "schoon schip maken": to make a clean sweep, to clean/clear things u ...
(Dutch for "clean ship") in 1963. Another early system was
FORMAC __NOTOC__ FORMAC, the FORmula MAnipulation Compiler, was the first computer algebra system to have significant use. It was developed by Jean E. Sammet and her team, as an extension of FORTRAN IV. The compiler was implemented as a preprocessor tak ...
. Using
Lisp A lisp is a speech impairment in which a person misarticulates sibilants (, , , , , , , ). These misarticulations often result in unclear speech. Types * A frontal lisp occurs when the tongue is placed anterior to the target. Interdental lisping ...
as the programming basis,
Carl Engelman Carl Engelman (October 21, 1929 – November 26, 1983) was an American computer scientist. Carl is best known as the creator of MATHLAB. He was employed by Mitre Corporation and Symbolics. He was a visiting professor at the University of Turin ...
created
MATHLAB MATHLAB is a computer algebra system created in 1964 by Carl Engelman at MITRE and written in Lisp. "MATHLAB 68" was introduced in 1967 and became rather popular in university environments running on DECs PDP-6 and PDP-10 under TOPS-10 or TE ...
in 1964 at
MITRE The mitre (Commonwealth English) (; Greek: μίτρα, "headband" or "turban") or miter (American English; see spelling differences), is a type of headgear now known as the traditional, ceremonial headdress of bishops and certain abbots in ...
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 SIMH is a free and open source, multi-platform multi-system emulator. It is maintained by Bob Supnik, a former DEC engineer and DEC vice president, and has been in development in one form or another since the 1960s. History SIMH was based on ...
emulations of the PDP-10. MATHLAB ("mathematical laboratory") should not be confused with
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementatio ...
("matrix laboratory"), which is a system for numerical computation built 15 years later at the
University of New Mexico The University of New Mexico (UNM; es, Universidad de Nuevo México) is a public research university in Albuquerque, New Mexico. Founded in 1889, it is the state's flagship academic institution and the largest by enrollment, with over 25,40 ...
. The first popular computer algebra systems were
muMATH muMATH is a computer algebra system (CAS), which was developed in the late 1970s and early eighties by Albert D. Rich and David Stoutemyer of Soft Warehouse in Honolulu, Hawaii. It was implemented in the muSIMP programming language which was buil ...
,
Reduce Reduction, reduced, or reduce may refer to: Science and technology Chemistry * Reduction (chemistry), part of a reduction-oxidation (redox) reaction in which atoms have their oxidation state changed. ** Organic redox reaction, a redox reacti ...
, Derive (based on muMATH), and
Macsyma Macsyma (; "Project MAC's SYmbolic MAnipulator") is one of the oldest general-purpose computer algebra systems still in wide use. It was originally developed from 1968 to 1982 at MIT's Project MAC. In 1982, Macsyma was licensed to Symbolics and ...
; a popular
copyleft Copyleft is the legal technique of granting certain freedoms over copies of copyrighted works with the requirement that the same rights be preserved in derivative works. In this sense, ''freedoms'' refers to the use of the work for any purpose, ...
version of Macsyma called Maxima is actively being maintained.
Reduce Reduction, reduced, or reduce may refer to: Science and technology Chemistry * Reduction (chemistry), part of a reduction-oxidation (redox) reaction in which atoms have their oxidation state changed. ** Organic redox reaction, a redox reacti ...
became free software in 2008. As of today, the most popular commercial systems are
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...
and
Maple ''Acer'' () is a genus of trees and shrubs commonly known as maples. The genus is placed in the family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated since http ...
, which are commonly used by research mathematicians, scientists, and engineers. Freely available alternatives include
SageMath SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, numerical analysis, n ...
(which can act as a front-end to several other free and nonfree CAS). In 1987,
Hewlett-Packard The Hewlett-Packard Company, commonly shortened to Hewlett-Packard ( ) or HP, was an American multinational information technology company headquartered in Palo Alto, California. HP developed and provided a wide variety of hardware components ...
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. In 1999, the independently developed CAS
Erable Erable is a computer algebra system (CAS) for a family of Hewlett-Packard graphing scientific calculators of the HP 40, 48 and HP 49/50 series. History Originally named ''ALGB'' in 1993, it was developed by the French mathematician for ...
for the HP 48 series became an officially integrated part of the firmware of the emerging HP 49/50 series, and a year later into the HP 40 series as well, whereas the
HP Prime The HP Prime Graphing Calculator is a graphing calculator introduced by Hewlett-Packard in 2013 and currently manufactured by HP Inc. It was designed with features resembling those of smartphones, such as a full-color touchscreen display and ...
adopted the Xcas system in 2013. The
Texas Instruments Texas Instruments Incorporated (TI) is an American technology company headquartered in Dallas, Texas, that designs and manufactures semiconductors and various integrated circuits, which it sells to electronics designers and manufacturers globall ...
company in 1995 released the
TI-92 The TI-92 series of graphing calculators are a line of calculators produced by Texas Instruments. They include: the TI-92 (1995), the TI-92 II (1996), the TI-92 Plus (1998, 1999) and the Voyage 200 (2002). The design of these relatively large ...
calculator with a CAS based on the software Derive; the TI-Nspire series replaced Derive in 2007. The
TI-89 series The TI-89 and the TI-89 Titanium are graphing calculators developed by Texas Instruments (TI). They are differentiated from most other TI graphing calculators by their computer algebra system, which allows symbolic manipulation of algeb ...
, first released in 1998, also contains a CAS.
Casio is a Japanese multinational electronics manufacturing corporation headquartered in Shibuya, Tokyo, Japan. Its products include calculators, mobile phones, digital cameras, electronic musical instruments, and analogue and digital watches. It ...
released their first CAS calculator with the CFX-9970G and succeeded it with the Algebra FX Series in 1999-2003 and the current ClassPad Series. More recently, computer algebra systems have been implemented using
artificial neural networks Artificial neural networks (ANNs), usually simply called neural networks (NNs) or neural nets, are computing systems inspired by the biological neural networks that constitute animal brains. An ANN is based on a collection of connected uni ...
.


Symbolic manipulations

The symbolic manipulations supported typically include: *simplification to a smaller expression or some standard form, including automatic simplification with assumptions and simplification with constraints *
substitution Substitution may refer to: Arts and media * Chord substitution, in music, swapping one chord for a related one within a chord progression * Substitution (poetry), a variation in poetic scansion * "Substitution" (song), a 2009 song by Silversun P ...
of symbols or numeric values for certain expressions *change of form of expressions: expanding products and powers, partial and full
factorization In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several ''factors'', usually smaller or simpler objects of the same kin ...
, rewriting as
partial fraction In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as ...
s,
constraint satisfaction In artificial intelligence and operations research, constraint satisfaction is the process of finding a solution through a set of constraints that impose conditions that the variables must satisfy. A solution is therefore a set of values for the ...
, rewriting
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
as exponentials, transforming logic expressions, etc. * partial and
total differentiation In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with resp ...
*some indefinite and definite integration (see
symbolic integration In calculus, symbolic integration is the problem of finding a formula for the antiderivative, or ''indefinite integral'', of a given function ''f''(''x''), i.e. to find a differentiable function ''F''(''x'') such that :\frac = f(x). This is als ...
), including multidimensional integrals *symbolic constrained and unconstrained global optimization * solution of linear and some non-linear equations over various domains *solution of some differential and
difference equation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
s *taking some limits *integral transforms * series operations such as expansion, summation and products *matrix operations including
products Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Prod ...
, inverses, etc. * statistical computation *
theorem proving Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a ma ...
and
verification Verify or verification may refer to: General * Verification and validation, in engineering or quality management systems, is the act of reviewing, inspecting or testing, in order to establish and document that a product, service or system meets ...
which is very useful in the area of
experimental mathematics Experimental mathematics is an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns. It has been defined as "that branch of mathematics that concerns itself ultimately with t ...
* optimized code generation In the above, the word ''some'' indicates that the operation cannot always be performed.


Additional capabilities

Many also include: *a
programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming ...
, allowing users to implement their own algorithms *
arbitrary-precision In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are li ...
numeric operations *exact integer arithmetic and number theory functionality * Editing of mathematical expressions in two-dimensional form *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 In computer programming, a string is traditionally a sequence of characters, either as a literal constant or as some kind of variable. The latter may allow its elements to be mutated and the length changed, or it may be fixed (after creation) ...
such as matching and
searching Searching or search may refer to: Computing technology * Search algorithm, including keyword search ** :Search algorithms * Search and optimization for problem solving in artificial intelligence * Search engine technology, software for findi ...
*add-ons for use in
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical ...
such as physics,
bioinformatics Bioinformatics () is an interdisciplinary field that develops methods and software tools for understanding biological data, in particular when the data sets are large and complex. As an interdisciplinary field of science, bioinformatics combine ...
,
computational chemistry Computational chemistry is a branch of chemistry that uses computer simulation to assist in solving chemical problems. It uses methods of theoretical chemistry, incorporated into computer programs, to calculate the structures and properties of m ...
and packages for physical computation *solvers for
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s Some include: *
graphic Graphics () are visual images or designs on some surface, such as a wall, canvas, screen, paper, or stone, to inform, illustrate, or entertain. In contemporary usage, it includes a pictorial representation of data, as in design and manufacture, ...
production and editing such as
computer-generated imagery Computer-generated imagery (CGI) is the use of computer graphics to create or contribute to images in art, printed media, video games, simulators, and visual effects in films, television programs, shorts, commercials, and videos. The image ...
and
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, d ...
as
image processing An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
*
sound synthesis A synthesizer (also spelled synthesiser) is an electronic musical instrument that generates audio signals. Synthesizers typically create sounds by generating waveforms through methods including subtractive synthesis, additive synthesis and ...
Some computer algebra systems focus on specialized disciplines; these are typically developed in academia and are free. They can be inefficient for numeric operations as compared to numeric systems.


Types of expressions

The expressions manipulated by the CAS typically include
polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
s in multiple variables; standard functions of expressions (
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is op ...
,
exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Expo ...
, etc.); various special functions ( Γ, ζ, erf,
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
s, etc.); arbitrary functions of expressions; optimization; derivatives, integrals, simplifications, sums, and products of expressions; truncated series with expressions as coefficients,
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
of expressions, and so on. Numeric domains supported typically include floating-point representation of real numbers,
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
s (of unbounded size),
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
(floating-point representation), interval representation of reals,
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
(exact representation) and
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the p ...
s.


Use in education

There have been many advocates for increasing the use of computer algebra systems in primary and secondary-school classrooms. The primary reason for such advocacy is that computer algebra systems represent real-world math more than do paper-and-pencil or hand calculator based mathematics. This push for increasing computer usage in mathematics classrooms has been supported by some boards of education. It has even been mandated in the curriculum of some regions. Computer algebra systems have been extensively used in higher education. Many universities offer either specific courses on developing their use, or they implicitly expect students to use them for their course work. The companies that develop computer algebra systems have pushed to increase their prevalence among university and college programs. CAS-equipped calculators are not permitted on the ACT, the
PLAN A plan is typically any diagram or list of steps with details of timing and resources, used to achieve an objective to do something. It is commonly understood as a temporal set of intended actions through which one expects to achieve a goal. ...
, and in some classroomsACT's CAAP Tests: Use of Calculators on the CAAP Mathematics Test
though it may be permitted on all of
College Board The College Board is an American nonprofit organization that was formed in December 1899 as the College Entrance Examination Board (CEEB) to expand access to higher education. While the College Board is not an association of colleges, it runs ...
's calculator-permitted tests, including the
SAT The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and scoring have changed several times; originally called the Scholastic Aptitude Test, it was later called the Schol ...
, some
SAT Subject Tests SAT Subject Tests were 20 multiple-choice standardized tests given by the College Board on individual subjects, typically taken to improve a student's credentials for college admissions in the United States. On January 19, 2021, the College Boar ...
and the
AP Calculus Advanced Placement (AP) Calculus (also known as AP Calc, Calc AB / Calc BC or simply AB / BC) is a set of two distinct Advanced Placement calculus courses and exams offered by the American nonprofit organization College Board. AP Calculus AB cover ...
, Chemistry,
Physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relat ...
, and
Statistics Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industri ...
exams.


Mathematics used in computer algebra systems

*
Knuth–Bendix completion algorithm The Knuth–Bendix completion algorithm (named after Donald Knuth and Peter Bendix) is a semi-decision algorithm for transforming a set of equations (over terms) into a confluent term rewriting system. When the algorithm succeeds, it effectively ...
*
Root-finding algorithm In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbe ...
s *
Symbolic integration In calculus, symbolic integration is the problem of finding a formula for the antiderivative, or ''indefinite integral'', of a given function ''f''(''x''), i.e. to find a differentiable function ''F''(''x'') such that :\frac = f(x). This is als ...
via e.g.
Risch algorithm In symbolic computation, the Risch algorithm is a method of indefinite integration used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra w ...
or Risch–Norman algorithm * Hypergeometric summation via e.g.
Gosper's algorithm In mathematics, Gosper's algorithm, due to Bill Gosper, is a procedure for finding sums of hypergeometric terms that are themselves hypergeometric terms. That is: suppose one has ''a''(1) + ... + ''a''(''n'') = ''S''(''n'')&n ...
* Limit computation via e.g. Gruntz's algorithm *
Polynomial factorization In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same doma ...
via e.g., over finite fields, Berlekamp's algorithm or Cantor–Zassenhaus algorithm. *
Greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
via e.g.
Euclidean algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an ef ...
*
Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
*
Gröbner basis In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field . A Gröbne ...
via e.g.
Buchberger's algorithm In the theory of multivariate polynomials, Buchberger's algorithm is a method for transforming a given set of polynomials into a Gröbner basis, which is another set of polynomials that have the same common zeros and are more convenient for extract ...
; generalization of Euclidean algorithm and Gaussian elimination *
Padé approximant In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is a ...
*
Schwartz–Zippel lemma In mathematics, the Schwartz–Zippel lemma (also called the DeMillo–Lipton–Schwartz–Zippel lemma) is a tool commonly used in probabilistic polynomial identity testing, i.e. in the problem of determining whether a given multivariate polynomia ...
and testing polynomial identities *
Chinese remainder theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of the ...
*
Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a ...
s *
Quantifier elimination Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement "\exists x such that \ldots" can be viewed as a question "When is there an x such t ...
over real numbers via e.g. Tarski's method/
Cylindrical algebraic decomposition In mathematics, cylindrical algebraic decomposition (CAD) is a notion, and an algorithm to compute it, that are fundamental for computer algebra and real algebraic geometry. Given a set ''S'' of polynomials in R''n'', a cylindrical algebraic decomp ...
* Landau's algorithm (nested radicals) * Derivatives of
elementary function In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exp ...
s and
special functions Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by ...
. (e.g. See derivatives of the incomplete gamma function.) *
Cylindrical algebraic decomposition In mathematics, cylindrical algebraic decomposition (CAD) is a notion, and an algorithm to compute it, that are fundamental for computer algebra and real algebraic geometry. Given a set ''S'' of polynomials in R''n'', a cylindrical algebraic decomp ...


See also

*
List of computer algebra systems The following tables provide a comparison of computer algebra systems (CAS). A CAS is a package comprising a set of algorithms for performing symbolic manipulations on algebraic objects, a language to implement them, and an environment in which to ...
*
Scientific computation Computational science, also known as scientific computing or scientific computation (SC), is a field in mathematics that uses advanced computing capabilities to understand and solve complex problems. It is an area of science that spans many dis ...
*
Statistical package Statistical software are specialized computer programs for analysis in statistics and econometrics. Open-source * ADaMSoft – a generalized statistical software with data mining algorithms and methods for data management * ADMB – a softwar ...
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Automated theorem proving Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a m ...
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Algebraic modeling language Algebraic modeling languages (AML) are high-level computer programming languages for describing and solving high complexity problems for large scale mathematical computation (i.e. large scale optimization type problems). One particular advantage of ...
* Constraint-logic programming *
Satisfiability modulo theories In computer science and mathematical logic, satisfiability modulo theories (SMT) is the problem of determining whether a mathematical formula is satisfiable. It generalizes the Boolean satisfiability problem (SAT) to more complex formulas involv ...


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External links


Curriculum and Assessment in an Age of Computer Algebra Systems
- From the
Education Resources Information Center The Education Resources Information Center (ERIC) is an online digital library of education research and information. ERIC is sponsored by the Institute of Education Sciences of the United States Department of Education. Description The missio ...
Clearinghouse for Science, Mathematics, and Environmental Education,
Columbus, Ohio Columbus () is the state capital and the most populous city in the U.S. state of Ohio. With a 2020 census population of 905,748, it is the 14th-most populous city in the U.S., the second-most populous city in the Midwest, after Chicago, and ...
. *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 website

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