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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Computer algebra system</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">"Symbolic algebra" redirects here. For the algebra of logic, see <a href="Symbolical_algebra" class="mw-redirect" title="Symbolical algebra">Symbolical algebra</a>.</div>

<p>A <b>computer algebra system</b> (<b>CAS</b>) or <b>symbolic algebra system</b> (<b>SAS</b>) is any <a href="Mathematical_software" title="Mathematical software">mathematical software</a> with the ability to manipulate <a href="Mathematical_expressions" class="mw-redirect" title="Mathematical expressions">mathematical expressions</a> in a way similar to the traditional manual computations of <a href="Mathematician" title="Mathematician">mathematicians</a> and <a href="Scientist" title="Scientist">scientists</a>. The development of the computer algebra systems in the second half of the 20th century is part of the discipline of "<a href="Computer_algebra" title="Computer algebra">computer algebra</a>" or "symbolic computation", which has spurred work in <a href="Algorithm" title="Algorithm">algorithms</a> over <a href="Mathematical_object" title="Mathematical object">mathematical objects</a> such as <a href="Polynomial" title="Polynomial">polynomials</a>.

</p><p>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 <a href="Number_theory" title="Number theory">number theory</a>, <a href="Group_theory" title="Group theory">group theory</a>, or teaching of <a href="Elementary_mathematics" title="Elementary mathematics">elementary mathematics</a>.

</p><p>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:

</p>

<ul><li>a <a href="User_interface" title="User interface">user interface</a> allowing a user to enter and display mathematical formulas, typically from a keyboard, menu selections, mouse or stylus.</li>

<li>a <a href="Programming_language" title="Programming language">programming language</a> and an <a href="Interpreter_(computing)" title="Interpreter (computing)">interpreter</a> (the result of a computation commonly has an unpredictable form and an unpredictable size; therefore user intervention is frequently needed),</li>

<li>a <a href="Symbolic_computation" class="mw-redirect" title="Symbolic computation">simplifier</a>, which is a <a href="Rewrite_system" class="mw-redirect" title="Rewrite system">rewrite system</a> for simplifying mathematics formulas,</li>

<li>a <a href="Memory_management" title="Memory management">memory manager</a>, including a <a href="Garbage_collector_(computing)" class="mw-redirect" title="Garbage collector (computing)">garbage collector</a>, needed by the huge size of the intermediate data, which may appear during a computation,</li>

<li>an <a href="Arbitrary-precision_arithmetic" title="Arbitrary-precision arithmetic">arbitrary-precision arithmetic</a>, needed by the huge size of the integers that may occur,</li>

<li>a large library of mathematical <a href="Algorithm" title="Algorithm">algorithms</a> and <a href="Special_functions" title="Special functions">special functions</a>.</li></ul>

<p>The library must not only provide for the needs of the users, but also the needs of the simplifier. For example, the computation of <a href="Polynomial_greatest_common_divisor" title="Polynomial greatest common divisor">polynomial greatest common divisors</a> is systematically used for the simplification of expressions involving fractions.

</p><p>This large amount of required computer capabilities explains the small number of general-purpose computer algebra systems. Significant systems include <a href="Axiom_(computer_algebra_system)" title="Axiom (computer algebra system)">Axiom</a>, <a href="GAP_(computer_algebra_system)" title="GAP (computer algebra system)">GAP</a>, <a href="Maxima_(software)" title="Maxima (software)">Maxima</a>, <a href="Magma_(computer_algebra_system)" title="Magma (computer algebra system)">Magma</a>, <a href="Maple_(software)" title="Maple (software)">Maple</a>, <a href="Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a>, and <a href="SageMath" title="SageMath">SageMath</a>.

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<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<p>In the 1950s, while computers were mainly used for numerical computations, there were some research projects into using them for symbolic manipulation. 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 <a href="Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a>.

</p><p>A prime example for the first development was the pioneering work conducted by the later Nobel Prize laureate in physics <a href="Martinus_Veltman" class="mw-redirect" title="Martinus Veltman">Martinus Veltman</a>, who designed a program for symbolic mathematics, especially high-energy physics, called <a href="Schoonschip" title="Schoonschip">Schoonschip</a> (Dutch for "clean ship") in 1963. Other early systems include <a href="FORMAC_(programming_language)" class="mw-redirect" title="FORMAC (programming language)">FORMAC</a>.

</p><p>Using <a href="Lisp_(programming_language)" title="Lisp (programming language)">Lisp</a> as the programming basis, <a href="Carl_Engelman" title="Carl Engelman">Carl Engelman</a> created <a href="MATHLAB" title="MATHLAB">MATHLAB</a> in 1964 at <a href="MITRE" class="mw-redirect" title="MITRE">MITRE</a> 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 <a href="SIMH" title="SIMH">SIMH</a> emulations of the PDP-10. MATHLAB ("<b>math</b>ematical <b>lab</b>oratory") should not be confused with <a href="MATLAB" title="MATLAB">MATLAB</a> ("<b>mat</b>rix <b>lab</b>oratory"), which is a system for numerical computation built 15 years later at the <a href="University_of_New_Mexico" title="University of New Mexico">University of New Mexico</a>.

</p><p>In 1987, <a href="Hewlett-Packard" title="Hewlett-Packard">Hewlett-Packard</a> introduced the first hand-held calculator CAS with the <a href="HP-28_series" title="HP-28 series">HP-28 series</a>.<sup id="cite_ref-Nelson_1-0" class="reference"><a href="#cite_note-Nelson-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Other early handheld calculators with symbolic algebra capabilities included the <a href="Texas_Instruments" title="Texas Instruments">Texas Instruments</a> <a href="TI-89_series" title="TI-89 series">TI-89 series</a> and <a href="TI-92" class="mw-redirect" title="TI-92">TI-92</a> calculator, and the <a href="Casio" title="Casio">Casio</a> <a href="CFX-9970G" class="mw-redirect" title="CFX-9970G">CFX-9970G</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>The first popular computer algebra systems were <a href="MuMATH" title="MuMATH">muMATH</a>, <a href="Reduce_computer_algebra_system" class="mw-redirect" title="Reduce computer algebra system">Reduce</a>, <a href="Derive_(computer_algebra_system)" title="Derive (computer algebra system)">Derive</a> (based on muMATH), and <a href="Macsyma" title="Macsyma">Macsyma</a>; a <a href="Copyleft" title="Copyleft">copyleft</a> version of Macsyma is called <a href="Maxima_(software)" title="Maxima (software)">Maxima</a>. <a href="Reduce_(computer_algebra_system)" title="Reduce (computer algebra system)">Reduce</a> became free software in 2008.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Commercial systems include <a href="Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and <a href="Maple_(software)" title="Maple (software)">Maple</a>, which are commonly used by research mathematicians, scientists, and engineers. Freely available alternatives include <a href="SageMath" title="SageMath">SageMath</a> (which can act as a <a href="Front_and_back_ends" class="mw-redirect" title="Front and back ends">front-end</a> to several other free and nonfree CAS). Other significant systems include <a href="Axiom_(computer_algebra_system)" title="Axiom (computer algebra system)">Axiom</a>, <a href="GAP_(computer_algebra_system)" title="GAP (computer algebra system)">GAP</a>, <a href="Maxima_(software)" title="Maxima (software)">Maxima</a> and <a href="Magma_(computer_algebra_system)" title="Magma (computer algebra system)">Magma</a>.

</p><p>The movement to web-based applications in the early 2000s saw the release of <a href="WolframAlpha" title="WolframAlpha">WolframAlpha</a>, an online search engine and CAS which includes the capabilities of <a href="Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p><p>More recently, computer algebra systems have been implemented using <a href="Artificial_neural_networks" class="mw-redirect" title="Artificial neural networks">artificial neural networks</a>, though as of 2020 they are not commercially available.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Symbolic_manipulations">Symbolic manipulations</h2></div>

<p>The symbolic manipulations supported typically include:

</p>

<ul><li>simplification to a smaller expression or some <a href="Canonical_form" title="Canonical form">standard form</a>, including automatic simplification with assumptions and simplification with constraints</li>

<li><a href="Substitution_(algebra)" class="mw-redirect" title="Substitution (algebra)">substitution</a> of symbols or numeric values for certain expressions</li>

<li>change of form of expressions: expanding products and powers, partial and full <a href="Factorization" title="Factorization">factorization</a>, rewriting as <a href="Partial_fraction" class="mw-redirect" title="Partial fraction">partial fractions</a>, <a href="Constraint_satisfaction" title="Constraint satisfaction">constraint satisfaction</a>, rewriting <a href="Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a> as exponentials, transforming logic expressions, etc.</li>

<li><a href="Partial_differentiation" class="mw-redirect" title="Partial differentiation">partial</a> and <a href="Total_differentiation" class="mw-redirect" title="Total differentiation">total differentiation</a></li>

<li>some <a href="Antidifferentiation" class="mw-redirect" title="Antidifferentiation">indefinite</a> and <a href="Integral" title="Integral">definite integration</a> (see <a href="Symbolic_integration" title="Symbolic integration">symbolic integration</a>), including multidimensional integrals</li>

<li>symbolic constrained and unconstrained global optimization</li>

<li><a href="Closed-form_solution" class="mw-redirect" title="Closed-form solution">solution</a> of linear and some non-linear equations over various domains</li>

<li>solution of some <a href="Differential_equation" title="Differential equation">differential</a> and <a href="Difference_equation" class="mw-redirect" title="Difference equation">difference equations</a></li>

<li>taking some <a href="Limit_of_a_function" title="Limit of a function">limits</a></li>

<li>integral transforms</li>

<li><a href="Mathematical_series" class="mw-redirect" title="Mathematical series">series</a> operations such as expansion, summation and products</li>

<li>matrix operations including <a href="Direct_product_(Matrix)" class="mw-redirect" title="Direct product (Matrix)">products</a>, <a href="Matrix_inversion" class="mw-redirect" title="Matrix inversion">inverses</a>, etc.</li>

<li><a href="List_of_statistical_packages" class="mw-redirect" title="List of statistical packages">statistical computation</a></li>

<li><a href="Theorem_proving" class="mw-redirect" title="Theorem proving">theorem proving</a> and <a href="Proof_verification" class="mw-redirect" title="Proof verification">verification</a> which is very useful in the area of <a href="Experimental_mathematics" title="Experimental mathematics">experimental mathematics</a></li>

<li><a href="Automatic_programming#Implementations" title="Automatic programming">optimized code generation</a></li></ul>

<p>In the above, the word <i>some</i> indicates that the operation cannot always be performed.

</p>

<div class="mw-heading mw-heading2"><h2 id="Additional_capabilities">Additional capabilities</h2></div>

<p>Many also include:

</p>

<ul><li>a <a href="Programming_language" title="Programming language">programming language</a>, allowing users to implement their own algorithms</li>

<li><a href="Arbitrary-precision" class="mw-redirect" title="Arbitrary-precision">arbitrary-precision</a> numeric operations</li>

<li>exact integer arithmetic and number theory functionality</li>

<li><a href="Formula_editor" title="Formula editor">Editing of mathematical expressions</a> in two-dimensional form</li>

<li>plotting <a href="Graph_of_a_function" title="Graph of a function">graphs and parametric plots</a> of functions in two and three dimensions, and animating them</li>

<li>drawing charts and diagrams</li>

<li><a href="Application_programming_interface" class="mw-redirect" title="Application programming interface">APIs</a> for linking it on an external program such as a database, or using in a programming language to use the computer algebra system</li>

<li><a href="String_manipulation" class="mw-redirect" title="String manipulation">string manipulation</a> such as <a href="String_matching" class="mw-redirect" title="String matching">matching</a> and <a href="String_searching" class="mw-redirect" title="String searching">searching</a></li>

<li>add-ons for use in <a href="Applied_mathematics" title="Applied mathematics">applied mathematics</a> such as physics, <a href="Bioinformatics" title="Bioinformatics">bioinformatics</a>, <a href="Computational_chemistry" title="Computational chemistry">computational chemistry</a> and packages for <a href="Computational_physics" title="Computational physics">physical computation</a><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li>

<li>solvers for <a href="Differential_equation" title="Differential equation">differential equations</a><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li></ul>

<p>Some include:

</p>

<ul><li><a href="Computer_graphics" title="Computer graphics">graphic</a> production and editing such as <a href="Computer-generated_imagery" title="Computer-generated imagery">computer-generated imagery</a> and <a href="Signal_processing" title="Signal processing">signal processing</a> as <a href="Image_processing" class="mw-redirect" title="Image processing">image processing</a></li>

<li><a href="Sound_synthesis" class="mw-redirect" title="Sound synthesis">sound synthesis</a></li></ul>

<p>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 <a href="Comparison_of_numerical-analysis_software" title="Comparison of numerical-analysis software">numeric systems</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Types_of_expressions">Types of expressions</h2></div>

<p>The expressions manipulated by the CAS typically include <a href="Polynomial" title="Polynomial">polynomials</a> in multiple variables; standard functions of expressions (<a href="Trigonometric_functions" title="Trigonometric functions">sine</a>, <a href="Exponential_function" title="Exponential function">exponential</a>, etc.); various special functions (<a href="Gamma_function" title="Gamma function">Γ</a>, <a href="Riemann_zeta_function" title="Riemann zeta function">ζ</a>, <a href="Error_function" title="Error function">erf</a>, <a href="Bessel_function" title="Bessel function">Bessel functions</a>, etc.); arbitrary functions of expressions; optimization; derivatives, integrals, simplifications, sums, and products of expressions; truncated <a href="Series_(mathematics)" title="Series (mathematics)">series</a> with expressions as coefficients, <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> of expressions, and so on. Numeric domains supported typically include <a href="Floating-point_arithmetic" title="Floating-point arithmetic">floating-point representation of real numbers</a>, <a href="Integer" title="Integer">integers</a> (of unbounded size), <a href="Complex_number" title="Complex number">complex</a> (floating-point representation), <a href="Interval_arithmetic" title="Interval arithmetic">interval representation of reals</a>, <a href="Rational_number" title="Rational number">rational number</a> (exact representation) and <a href="Algebraic_number" title="Algebraic number">algebraic numbers</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Use_in_education">Use in education</h2></div>

<p>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.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>

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.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>

</p><p>Computer algebra systems have been extensively used in higher education.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> 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.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>

</p><p>CAS-equipped calculators are not permitted on the <a href="ACT_(test)" title="ACT (test)">ACT</a>, the <a href="PLAN_(test)" title="PLAN (test)">PLAN</a>, and in some classrooms<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> though it may be permitted on all of <a href="College_Board" title="College Board">College Board</a>'s calculator-permitted tests, including the <a href="SAT" title="SAT">SAT</a>, some <a href="SAT_Subject_Tests" title="SAT Subject Tests">SAT Subject Tests</a> and the <a href="AP_Calculus" title="AP Calculus">AP Calculus</a>, <a href="AP_Chemistry" title="AP Chemistry">Chemistry</a>, <a href="AP_Physics" title="AP Physics">Physics</a>, and <a href="AP_Statistics" title="AP Statistics">Statistics</a> exams.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Mathematics_used_in_computer_algebra_systems">Mathematics used in computer algebra systems</h2></div>

<ul><li><a href="Knuth%E2%80%93Bendix_completion_algorithm" title="Knuth–Bendix completion algorithm">Knuth–Bendix completion algorithm</a><sup id="cite_ref-BuchbergerCollins2013_20-0" class="reference"><a href="#cite_note-BuchbergerCollins2013-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></li>

<li><a href="Root-finding_algorithm" title="Root-finding algorithm">Root-finding algorithms</a><sup id="cite_ref-BuchbergerCollins2013_20-1" class="reference"><a href="#cite_note-BuchbergerCollins2013-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></li>

<li><a href="Symbolic_integration" title="Symbolic integration">Symbolic integration</a> via e.g. <a href="Risch_algorithm" title="Risch algorithm">Risch algorithm</a> or <a href="Risch%E2%80%93Norman_algorithm" class="mw-redirect" title="Risch–Norman algorithm">Risch–Norman algorithm</a></li>

<li><a href="Hypergeometric_summation" class="mw-redirect" title="Hypergeometric summation">Hypergeometric summation</a> via e.g. <a href="Gosper's_algorithm" title="Gosper's algorithm">Gosper's algorithm</a></li>

<li><a href="Limit_(mathematics)" title="Limit (mathematics)">Limit computation</a> via e.g. Gruntz's algorithm</li>

<li><a href="Polynomial_factorization" class="mw-redirect" title="Polynomial factorization">Polynomial factorization</a> via e.g., over finite fields,<sup id="cite_ref-GathenGerhard2013_21-0" class="reference"><a href="#cite_note-GathenGerhard2013-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> <a href="Berlekamp's_algorithm" title="Berlekamp's algorithm">Berlekamp's algorithm</a> or <a href="Cantor%E2%80%93Zassenhaus_algorithm" title="Cantor–Zassenhaus algorithm">Cantor–Zassenhaus algorithm</a>.</li>

<li><a href="Greatest_common_divisor" title="Greatest common divisor">Greatest common divisor</a> via e.g. <a href="Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a></li>

<li><a href="Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a><sup id="cite_ref-GeddesCzapor2007_22-0" class="reference"><a href="#cite_note-GeddesCzapor2007-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></li>

<li><a href="Gr%C3%B6bner_basis" title="Gröbner basis">Gröbner basis</a> via e.g. <a href="Buchberger's_algorithm" title="Buchberger's algorithm">Buchberger's algorithm</a>; generalization of Euclidean algorithm and Gaussian elimination</li>

<li><a href="Pad%C3%A9_approximant" title="Padé approximant">Padé approximant</a></li>

<li><a href="Schwartz%E2%80%93Zippel_lemma" title="Schwartz–Zippel lemma">Schwartz–Zippel lemma</a> and testing polynomial identities</li>

<li><a href="Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a></li>

<li><a href="Diophantine_equation" title="Diophantine equation">Diophantine equations</a></li>

<li><a href="Landau's_algorithm" class="mw-redirect" title="Landau's algorithm">Landau's algorithm</a> (nested radicals)</li>

<li>Derivatives of <a href="Elementary_function" title="Elementary function">elementary functions</a> and <a href="Special_functions" title="Special functions">special functions</a>. (e.g. See <a href="Derivatives_of_the_incomplete_gamma_function" class="mw-redirect" title="Derivatives of the incomplete gamma function">derivatives of the incomplete gamma function</a>.)</li>

<li><a href="Cylindrical_algebraic_decomposition" title="Cylindrical algebraic decomposition">Cylindrical algebraic decomposition</a></li>

<li><a href="Quantifier_elimination" title="Quantifier elimination">Quantifier elimination</a> over real numbers via cylindrical algebraic decomposition</li></ul>

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<ul><li><a href="List_of_computer_algebra_systems" title="List of computer algebra systems">List of computer algebra systems</a></li>

<li><a href="Scientific_computation" class="mw-redirect" title="Scientific computation">Scientific computation</a></li>

<li><a href="Statistical_package" class="mw-redirect" title="Statistical package">Statistical package</a></li>

<li><a href="Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li>

<li><a href="Algebraic_modeling_language" title="Algebraic modeling language">Algebraic modeling language</a></li>

<li><a href="Constraint-logic_programming" class="mw-redirect" title="Constraint-logic programming">Constraint-logic programming</a></li>

<li><a href="Satisfiability_modulo_theories" title="Satisfiability modulo theories">Satisfiability modulo theories</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFNelson" class="citation web cs1">Nelson, Richard. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100703031935/http://h20331.www2.hp.com/Hpsub/cache/392617-0-0-225-121.html">"Hewlett-Packard Calculator Firsts"</a>. Hewlett-Packard. Archived from <a rel="nofollow" class="external text" href="http://h20331.www2.hp.com/Hpsub/cache/392617-0-0-225-121.html">the original</a> on 2010-07-03.</cite></span>

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<li id="cite_note-GathenGerhard2013-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-GathenGerhard2013_21-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFJoachim_von_zur_GathenJürgen_Gerhard2013" class="citation book cs1">Joachim von zur Gathen; Jürgen Gerhard (2013-04-25). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7fE9baKyqSEC&q=%22polynomial+factorization%22+%22finite+field%22"><i>Modern Computer Algebra</i></a>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-107-03903-2</bdi>.</cite></span>

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<li id="cite_note-GeddesCzapor2007-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-GeddesCzapor2007_22-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFKeith_O._GeddesStephen_R._CzaporGeorge_Labahn2007" class="citation book cs1">Keith O. Geddes; Stephen R. Czapor; George Labahn (2007-06-30). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9fOUwkkRxT4C"><i>Algorithms for Computer Algebra</i></a>. Springer Science & Business Media. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-585-33247-5</bdi>.</cite></span>

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</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.ericdigests.org/2003-1/age.htm">Curriculum and Assessment in an Age of Computer Algebra Systems</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091201030924/http://www.ericdigests.org/2003-1/age.htm">Archived</a> 2009-12-01 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a> - From the <a href="Education_Resources_Information_Center" title="Education Resources Information Center">Education Resources Information Center</a> Clearinghouse for Science, Mathematics, and Environmental Education, <a href="Columbus%2C_Ohio" title="Columbus, Ohio">Columbus, Ohio</a>.</li>

<li>Richard J. Fateman. "Essays in algebraic simplification." Technical report MIT-LCS-TR-095, 1972. <i>(Of historical interest in showing the direction of research in computer algebra. At the MIT LCS website: <a rel="nofollow" class="external autonumber" href="https://web.archive.org/web/20060917023934/http://www.lcs.mit.edu/publications/specpub.php?id=663">[1]</a>)</i></li></ul>

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</style><div id="Computer_algebra_systems377" style="font-size:114%;margin:0 4em"></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Open-source_software" title="Open-source software">Open-source</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Axiom_(computer_algebra_system)" title="Axiom (computer algebra system)">Axiom</a></li>

<li><a href="Cadabra_(computer_program)" title="Cadabra (computer program)">Cadabra</a></li>

<li><a href="CoCoA" title="CoCoA">CoCoA</a></li>

<li><a href="Fermat_(computer_algebra_system)" title="Fermat (computer algebra system)">Fermat</a></li>

<li><a href="FriCAS" title="FriCAS">FriCAS</a></li>

<li><a href="GiNaC" title="GiNaC">GiNaC</a></li>

<li><a href="Macaulay2" title="Macaulay2">Macaulay2</a></li>

<li><a href="Maxima_(software)" title="Maxima (software)">Maxima</a></li>

<li><a href="Normaliz" title="Normaliz">Normaliz</a></li>

<li><a href="Reduce_(computer_algebra_system)" title="Reduce (computer algebra system)">Reduce</a></li>

<li><a href="SageMath" title="SageMath">SageMath</a></li>

<li><a href="Singular_(software)" title="Singular (software)">Singular</a></li>

<li><a href="SymPy" title="SymPy">SymPy</a></li>

<li><a href="Yacas" title="Yacas">Yacas</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Proprietary_software" title="Proprietary software">Proprietary</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Casio_ClassPad_300" title="Casio ClassPad 300">ClassPad Manager</a></li>

<li><a href="Engineering_Equation_Solver" title="Engineering Equation Solver">Engineering Equation Solver</a></li>

<li><a href="Magma_(computer_algebra_system)" title="Magma (computer algebra system)">Magma</a></li>

<li><a href="Maple_(software)" title="Maple (software)">Maple</a></li>

<li><a href="Mathcad" title="Mathcad">Mathcad</a></li>

<li><a href="MuPAD" title="MuPAD">muPAD</a> (<a href="MATLAB" title="MATLAB">MATLAB</a> symbolic math toolbox)</li>

<li><a href="SMath_Studio" title="SMath Studio">SMath Studio</a></li>

<li><a href="TI_InterActive!" title="TI InterActive!">TI InterActive!</a></li>

<li><a href="Wolfram_(software)" title="Wolfram (software)">Wolfram</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Discontinued</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cambridge_Algebra_System" title="Cambridge Algebra System">CAMAL</a></li>

<li><a href="Derive_(computer_algebra_system)" title="Derive (computer algebra system)">Derive</a></li>

<li><a href="Erable" title="Erable">Erable</a></li>

<li><a href="LiveMath" title="LiveMath">LiveMath</a></li>

<li><a href="Macsyma" title="Macsyma">Macsyma</a></li>

<li><a href="Mathomatic" title="Mathomatic">Mathomatic</a></li>

<li><a href="MuMATH" title="MuMATH">muMATH</a></li>

<li><a href="ALTRAN" title="ALTRAN">ALTRAN</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span><b>Category</b></li>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-11" href="https://en.wikipedia.org/wiki/?title=Computer_algebra_system&oldid=1300041643">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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