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vector-toc-level-2"> <a class="vector-toc-link" href="#Foundations_and_early_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Foundations and early applications</span> </div> </a> <ul id="toc-Foundations_and_early_applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Historic_problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Historic_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Historic problems</span> </div> </a> <ul id="toc-Historic_problems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Algorithms_used_in_computer_algebra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algorithms_used_in_computer_algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Algorithms used in computer algebra</span> </div> </a> <ul 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<div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Scientific area at the interface between computer science and mathematics</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Lead_extra_info plainlinks metadata ambox ambox-style ambox-lead_extra_info" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article's <b><a href="/wiki/Wikipedia:Manual_of_Style/Lead_section" title="Wikipedia:Manual of Style/Lead section">lead section</a> contains information that is not included elsewhere in the article</b>.<span class="hide-when-compact"> If the information is appropriate for the lead of the article, this information should also be included in the body of the article.</span> <span class="date-container"><i>(<span class="date">May 2020</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:RischIntegration.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/RischIntegration.PNG/500px-RischIntegration.PNG" decoding="async" width="500" height="450" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/0/09/RischIntegration.PNG 1.5x" data-file-width="738" data-file-height="664" /></a><figcaption><a href="/wiki/Symbolic_integration" title="Symbolic integration">Symbolic integration</a> of the <a href="/wiki/Algebraic_function" title="Algebraic function">algebraic function</a> <span class="texhtml"><i>f</i>(<i>x</i>) = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>x</i></span><span class="sr-only">/</span><span class="den"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>x</i><sup>4</sup> + 10<i>x</i><sup>2</sup> − 96<i>x</i> − 71</span></span></span></span>⁠</span></span> using the computer algebra system <i><a href="/wiki/Axiom_(computer_algebra_system)" title="Axiom (computer algebra system)">Axiom</a></i></figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and <a href="/wiki/Computer_science" title="Computer science">computer science</a>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <b>computer algebra</b>, also called <b>symbolic computation</b> or <b>algebraic computation</b>, is a scientific area that refers to the study and development of <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> and <a href="/wiki/Software" title="Software">software</a> for manipulating <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">mathematical expressions</a> and other <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a>. Although computer algebra could be considered a subfield of <a href="/wiki/Scientific_computing" class="mw-redirect" title="Scientific computing">scientific computing</a>, they are generally considered as distinct fields because scientific computing is usually based on <a href="/wiki/Numerical_computation" class="mw-redirect" title="Numerical computation">numerical computation</a> with approximate <a href="/wiki/Floating_point_number" class="mw-redirect" title="Floating point number">floating point numbers</a>, while symbolic computation emphasizes <i>exact</i> computation with expressions containing <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a> that have no given value and are manipulated as symbols. </p><p><a href="/wiki/Software" title="Software">Software</a> applications that perform symbolic calculations are called <i><a href="/wiki/Computer_algebra_system" title="Computer algebra system">computer algebra systems</a></i>, with the term <i>system</i> alluding to the complexity of the main applications that include, at least, a method to represent mathematical data in a computer, a user <a href="/wiki/Programming_language" title="Programming language">programming language</a> (usually different from the language used for the implementation), a dedicated memory manager, a <a href="/wiki/User_interface" title="User interface">user interface</a> for the input/output of mathematical expressions, a large set of <a href="/wiki/Function_(computer_science)" class="mw-redirect" title="Function (computer science)">routines</a> to perform usual operations, like simplification of expressions, <a href="/wiki/Differentiation_(mathematics)" class="mw-redirect" title="Differentiation (mathematics)">differentiation</a> using the <a href="/wiki/Chain_rule" title="Chain rule">chain rule</a>, <a href="/wiki/Polynomial_factorization" class="mw-redirect" title="Polynomial factorization">polynomial factorization</a>, <a href="/wiki/Indefinite_integration" class="mw-redirect" title="Indefinite integration">indefinite integration</a>, etc. </p><p>Computer algebra is widely used to experiment in mathematics and to design the formulas that are used in numerical programs. It is also used for complete scientific computations, when purely numerical methods fail, as in <a href="/wiki/Public_key_cryptography" class="mw-redirect" title="Public key cryptography">public key cryptography</a>, or for some <a href="/wiki/Non-linear" class="mw-redirect" title="Non-linear">non-linear</a> problems. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Terminology">Terminology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computer_algebra&action=edit&section=1" title="Edit section: Terminology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some authors distinguish <i>computer algebra</i> from <i>symbolic computation</i> using the latter name to refer to kinds of symbolic computation other than the computation with mathematical <a href="/wiki/Formula" title="Formula">formulas</a>. Some authors use <i>symbolic computation</i> for the computer science aspect of the subject and "computer algebra" for the mathematical aspect.<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> In some languages the name of the field is not a direct translation of its English name. Typically, it is called <i>calcul formel</i> in French, which means "formal computation". This name reflects the ties this field has with <a href="/wiki/Formal_methods" title="Formal methods">formal methods</a>. </p><p>Symbolic computation has also been referred to, in the past, as <i>symbolic manipulation</i>, <i>algebraic manipulation</i>, <i>symbolic processing</i>, <i>symbolic mathematics</i>, or <i>symbolic algebra</i>, but these terms, which also refer to non-computational manipulation, are no longer used in reference to computer algebra. </p> <div class="mw-heading mw-heading2"><h2 id="Scientific_community">Scientific community</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computer_algebra&action=edit&section=2" title="Edit section: Scientific community"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is no <a href="/wiki/Learned_society" title="Learned society">learned society</a> that is specific to computer algebra, but this function is assumed by the <a href="/wiki/Special_interest_group" title="Special interest group">special interest group</a> of the <a href="/wiki/Association_for_Computing_Machinery" title="Association for Computing Machinery">Association for Computing Machinery</a> named <a href="/wiki/SIGSAM" title="SIGSAM">SIGSAM</a> (Special Interest Group on Symbolic and Algebraic Manipulation).<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> </p><p>There are several annual conferences on computer algebra, the premier being <a href="/wiki/ISSAC" class="mw-redirect" title="ISSAC">ISSAC</a> (International Symposium on Symbolic and Algebraic Computation), which is regularly sponsored by SIGSAM.<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> </p><p>There are several journals specializing in computer algebra, the top one being <i><a href="/wiki/Journal_of_Symbolic_Computation" title="Journal of Symbolic Computation">Journal of Symbolic Computation</a></i> founded in 1985 by <a href="/wiki/Bruno_Buchberger" title="Bruno Buchberger">Bruno Buchberger</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> There are also several other journals that regularly publish articles in computer algebra.<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="Computer_science_aspects">Computer science aspects</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computer_algebra&action=edit&section=3" title="Edit section: Computer science aspects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Data_representation">Data representation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computer_algebra&action=edit&section=4" title="Edit section: Data representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As <a href="/wiki/Numerical_software" class="mw-redirect" title="Numerical software">numerical software</a> is highly efficient for approximate <a href="/wiki/Numerical_computation" class="mw-redirect" title="Numerical computation">numerical computation</a>, it is common, in computer algebra, to emphasize <i>exact</i> computation with exactly represented data. Such an exact representation implies that, even when the size of the output is small, the intermediate data generated during a computation may grow in an unpredictable way. This behavior is called <i>expression swell</i>.<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> To obviate this problem, various methods are used in the representation of the data, as well as in the algorithms that manipulate them.<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> </p> <div class="mw-heading mw-heading4"><h4 id="Numbers">Numbers</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computer_algebra&action=edit&section=5" title="Edit section: Numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The usual numbers systems used in <a href="/wiki/Numerical_computation" class="mw-redirect" title="Numerical computation">numerical computation</a> are <a href="/wiki/Floating_point" class="mw-redirect" title="Floating point">floating point</a> numbers and <a href="/wiki/Integers" class="mw-redirect" title="Integers">integers</a> of a fixed bounded size. Neither of these is convenient for computer algebra, due to expression swell.<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> </p><p>Therefore, the basic numbers used in computer algebra are the integers of the mathematicians, commonly represented by an unbounded signed sequence of <a href="/wiki/Numerical_digit" title="Numerical digit">digits</a> in some <a href="/wiki/Radix" title="Radix">base of numeration</a>, usually the largest base allowed by the <a href="/wiki/Machine_word" class="mw-redirect" title="Machine word">machine word</a>. These integers allow to define the <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>, which are <a href="/wiki/Irreducible_fraction" title="Irreducible fraction">irreducible fractions</a> of two integers. </p><p>Programming an efficient implementation of the arithmetic operations is a hard task. Therefore, most free <a href="/wiki/Computer_algebra_system" title="Computer algebra system">computer algebra systems</a> and some commercial ones such as <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a> and <a href="/wiki/Maple_(software)" title="Maple (software)">Maple</a>,<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> use the <a href="/wiki/GNU_Multiple_Precision_Arithmetic_Library" title="GNU Multiple Precision Arithmetic Library">GMP library</a>, which is thus a <i>de facto</i> standard. </p> <div class="mw-heading mw-heading4"><h4 id="Expressions">Expressions</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computer_algebra&action=edit&section=6" title="Edit section: Expressions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Cassidy.1985.015.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Cassidy.1985.015.gif/400px-Cassidy.1985.015.gif" decoding="async" width="400" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Cassidy.1985.015.gif/600px-Cassidy.1985.015.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Cassidy.1985.015.gif/800px-Cassidy.1985.015.gif 2x" data-file-width="1087" data-file-height="357" /></a><figcaption>Representation of the expression <span class="texhtml">(8 − 6) × (3 + 1)</span> as a <a href="/wiki/Lisp_(programming_language)" title="Lisp (programming language)">Lisp</a> tree, from a 1985 Master's Thesis<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></figcaption></figure> <p>Except for <a href="/wiki/Number" title="Number">numbers</a> and <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a>, every <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">mathematical expression</a> may be viewed as the symbol of an operator followed by a <a href="/wiki/Sequence" title="Sequence">sequence</a> of operands. In computer algebra software, the expressions are usually represented in this way. This representation is very flexible, and many things that seem not to be mathematical expressions at first glance, may be represented and manipulated as such. For example, an equation is an expression with "=" as an operator, a matrix may be represented as an expression with "matrix" as an operator and its rows as operands. </p><p>Even programs may be considered and represented as expressions with operator "procedure" and, at least, two operands, the list of parameters and the body, which is itself an expression with "body" as an operator and a sequence of instructions as operands. Conversely, any mathematical expression may be viewed as a program. For example, the expression <span class="texhtml"><i>a</i> + <i>b</i></span> may be viewed as a program for the addition, with <span class="texhtml"><i>a</i></span> and <span class="texhtml"><i>b</i></span> as parameters. Executing this program consists in <i>evaluating</i> the expression for given values of <span class="texhtml"><i>a</i></span> and <span class="texhtml"><i>b</i></span>; if they are not given any values, the result of the evaluation is simply its input. </p><p>This process of delayed evaluation is fundamental in computer algebra. For example, the operator "=" of the equations is also, in most computer algebra systems, the name of the program of the equality test: normally, the evaluation of an equation results in an equation, but, when an equality test is needed, either explicitly asked by the user through an "evaluation to a Boolean" command, or automatically started by the system in the case of a test inside a program, then the evaluation to a Boolean result is executed. </p><p>As the size of the operands of an expression is unpredictable and may change during a working session, the sequence of the operands is usually represented as a sequence of either <a href="/wiki/Pointer_(computer_programming)" title="Pointer (computer programming)">pointers</a> (like in <a href="/wiki/Macsyma" title="Macsyma">Macsyma</a>)<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> or entries in a <a href="/wiki/Hash_table" title="Hash table">hash table</a> (like in <a href="/wiki/Maple_(software)" title="Maple (software)">Maple</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Simplification">Simplification</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computer_algebra&action=edit&section=7" title="Edit section: Simplification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The raw application of the basic rules of <a href="/wiki/Derivative" title="Derivative">differentiation</a> with respect to <span class="texhtml"><i>x</i></span> on the expression <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c031175c5142c25946f910813bf41c034fed75c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.402ex; height:2.343ex;" alt="{\displaystyle a^{x}}"></span> gives the result </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot a^{x-1}\cdot 0+a^{x}\cdot \left(1\cdot \log a+x\cdot {\frac {0}{a}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mn>0</mn> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>⋅</mo> <mi>log</mi> <mo>⁡</mo> <mi>a</mi> <mo>+</mo> <mi>x</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>0</mn> <mi>a</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot a^{x-1}\cdot 0+a^{x}\cdot \left(1\cdot \log a+x\cdot {\frac {0}{a}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/948bb66437562354fcd159c28bf5826c270625ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.075ex; height:6.176ex;" alt="{\displaystyle x\cdot a^{x-1}\cdot 0+a^{x}\cdot \left(1\cdot \log a+x\cdot {\frac {0}{a}}\right).}"></span></dd></dl> <p>A simpler expression than this is generally desired, and simplification is needed when working with general expressions. </p><p>This simplification is normally done through <a href="/wiki/Rewriting" title="Rewriting">rewriting rules</a>.<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> There are several classes of rewriting rules to be considered. The simplest are rules that always reduce the size of the expression, like <span class="texhtml"><i>E</i> − <i>E</i> → 0</span> or <span class="texhtml">sin(0) → 0</span>. They are systematically applied in computer algebra systems. </p><p>A difficulty occurs with <a href="/wiki/Associative_operation" class="mw-redirect" title="Associative operation">associative operations</a> like addition and multiplication. The standard way to deal with associativity is to consider that addition and multiplication have an arbitrary number of operands, that is that <span class="texhtml"><i>a</i> + <i>b</i> + <i>c</i></span> is represented as <span class="texhtml">"+"(<i>a</i>, <i>b</i>, <i>c</i>)</span>. Thus <span class="texhtml"><i>a</i> + (<i>b</i> + <i>c</i>)</span> and <span class="texhtml">(<i>a</i> + <i>b</i>) + <i>c</i></span> are both simplified to <span class="texhtml">"+"(<i>a</i>, <i>b</i>, <i>c</i>)</span>, which is displayed <span class="texhtml"><i>a</i> + <i>b</i> + <i>c</i></span>. In the case of expressions such as <span class="texhtml"><i>a</i> − <i>b</i> + <i>c</i></span>, the simplest way is to systematically rewrite <span class="texhtml">−<i>E</i></span>, <span class="texhtml"><i>E</i> − <i>F</i></span>, <span class="texhtml"><i>E</i>/<i>F</i></span> as, respectively, <span class="texhtml">(−1)⋅<i>E</i></span>, <span class="texhtml"><i>E</i> + (−1)⋅<i>F</i></span>, <span class="texhtml"><i>E</i>⋅<i>F</i><sup>−1</sup></span>. In other words, in the internal representation of the expressions, there is no subtraction nor division nor unary minus, outside the representation of the numbers. </p><p>Another difficulty occurs with the <a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">commutativity</a> of addition and multiplication. The problem is to quickly recognize the <a href="/wiki/Like_terms" title="Like terms">like terms</a> in order to combine or cancel them. Testing every pair of terms is costly with very long sums and products. To address this, <a href="/wiki/Macsyma" title="Macsyma">Macsyma</a> sorts the operands of sums and products into an order that places like terms in consecutive places, allowing easy detection. In <a href="/wiki/Maple_(software)" title="Maple (software)">Maple</a>, a <a href="/wiki/Hash_function" title="Hash function">hash function</a> is designed for generating collisions when like terms are entered, allowing them to be combined as soon as they are introduced. This allows subexpressions that appear several times in a computation to be immediately recognized and stored only once. This saves memory and speeds up computation by avoiding repetition of the same operations on identical expressions. </p><p>Some rewriting rules sometimes increase and sometimes decrease the size of the expressions to which they are applied. This is the case of <a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">distributivity</a> or <a href="/wiki/Trigonometric_identity#Product-to-sum_and_sum-to-product_identities" class="mw-redirect" title="Trigonometric identity">trigonometric identities</a>. For example, the distributivity law allows rewriting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+1)^{4}\rightarrow x^{4}+4x^{3}+6x^{2}+4x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+1)^{4}\rightarrow x^{4}+4x^{3}+6x^{2}+4x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e03f87d64b16fd5c5ff1539bfa7d42f99e8567b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.303ex; height:3.176ex;" alt="{\displaystyle (x+1)^{4}\rightarrow x^{4}+4x^{3}+6x^{2}+4x+1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-1)(x^{4}+x^{3}+x^{2}+x+1)\rightarrow x^{5}-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-1)(x^{4}+x^{3}+x^{2}+x+1)\rightarrow x^{5}-1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5476ec745b358f64f0797a258873187c8ab6fe30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.604ex; height:3.176ex;" alt="{\displaystyle (x-1)(x^{4}+x^{3}+x^{2}+x+1)\rightarrow x^{5}-1.}"></span> As there is no way to make a good general choice of applying or not such a rewriting rule, such rewriting is done only when explicitly invoked by the user. For the distributivity, the computer function that applies this rewriting rule is typically called "expand". The reverse rewriting rule, called "factor", requires a non-trivial algorithm, which is thus a key function in computer algebra systems (see <a href="/wiki/Polynomial_factorization" class="mw-redirect" title="Polynomial factorization">Polynomial factorization</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_aspects">Mathematical aspects</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computer_algebra&action=edit&section=8" title="Edit section: Mathematical aspects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some fundamental mathematical questions arise when one wants to manipulate <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">mathematical expressions</a> in a computer. We consider mainly the case of the <a href="/wiki/Multivariate_polynomial" class="mw-redirect" title="Multivariate polynomial">multivariate</a> <a href="/wiki/Rational_fraction" class="mw-redirect" title="Rational fraction">rational fractions</a>. This is not a real restriction, because, as soon as the <a href="/wiki/Irrational_function" class="mw-redirect" title="Irrational function">irrational functions</a> appearing in an expression are simplified, they are usually considered as new indeterminates. For example, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\sin(x+y)^{2}+\log(z^{2}-5))^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\sin(x+y)^{2}+\log(z^{2}-5))^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13506c17bd4e9e56e73cd23225040361b99e0e1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.677ex; height:3.176ex;" alt="{\displaystyle (\sin(x+y)^{2}+\log(z^{2}-5))^{3}}"></span></dd></dl> <p>is viewed as a polynomial in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(x+y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(x+y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af84037b516d93ba4a9ff505fb4d6d187fba9e24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.99ex; height:2.843ex;" alt="{\displaystyle \sin(x+y)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log(z^{2}-5)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(z^{2}-5)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6db643b2482eca638e8bd23d1e0e665c7921c217" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.928ex; height:3.176ex;" alt="{\displaystyle \log(z^{2}-5)}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Equality">Equality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computer_algebra&action=edit&section=9" title="Edit section: Equality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are two notions of equality for <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">mathematical expressions</a>. <b>Syntactic equality</b> is the equality of their representation in a computer. This is easy to test in a program. <i>Semantic equality</i> is when two expressions represent the same mathematical object, as in </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+y)^{2}=x^{2}+2xy+y^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)^{2}=x^{2}+2xy+y^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ded19ab0bf6cc33e2cafa09691cbd0454f7a570" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.861ex; height:3.176ex;" alt="{\displaystyle (x+y)^{2}=x^{2}+2xy+y^{2}.}"></span></dd></dl> <p>It is known from <a href="/wiki/Richardson%27s_theorem" title="Richardson's theorem">Richardson's theorem</a> that there may not exist an algorithm that decides whether two expressions representing numbers are semantically equal if exponentials and logarithms are allowed in the expressions. Accordingly, (semantic) equality may be tested only on some classes of expressions such as the <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> and <a href="/wiki/Rational_fraction" class="mw-redirect" title="Rational fraction">rational fractions</a>. </p><p>To test the equality of two expressions, instead of designing specific algorithms, it is usual to put expressions in some <i><a href="/wiki/Canonical_form" title="Canonical form">canonical form</a></i> or to put their difference in a <i>normal form</i>, and to test the syntactic equality of the result. </p><p>In computer algebra, "canonical form" and "normal form" are not synonymous.<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> A <i>canonical form</i> is such that two expressions in canonical form are semantically equal if and only if they are syntactically equal, while a <i>normal form</i> is such that an expression in normal form is semantically zero only if it is syntactically zero. In other words, zero has a unique representation as an expression in normal form. </p><p>Normal forms are usually preferred in computer algebra for several reasons. Firstly, canonical forms may be more costly to compute than normal forms. For example, to put a polynomial in canonical form, one has to expand every product through <a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">distributivity</a>, while it is not necessary with a normal form (see below). Secondly, it may be the case, like for expressions involving radicals, that a canonical form, if it exists, depends on some arbitrary choices and that these choices may be different for two expressions that have been computed independently. This may make impracticable the use of a canonical form. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computer_algebra&action=edit&section=10" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Human-driven_computer_algebra">Human-driven computer algebra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computer_algebra&action=edit&section=11" title="Edit section: Human-driven computer algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Early computer algebra systems, such as the <a href="/wiki/ENIAC" title="ENIAC">ENIAC</a> at the <a href="/wiki/University_of_Pennsylvania" title="University of Pennsylvania">University of Pennsylvania</a>, relied on <a href="/wiki/Computer_(occupation)" title="Computer (occupation)">human computers</a> or programmers to reprogram it between calculations, manipulate its many physical modules (or panels), and feed its IBM card reader.<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> Female mathematicians handled the majority of ENIAC programming human-guided computation: <a href="/wiki/Jean_Bartik" title="Jean Bartik">Jean Jennings</a>, <a href="/wiki/Marlyn_Meltzer" title="Marlyn Meltzer">Marlyn Wescoff</a>, <a href="/wiki/Ruth_Teitelbaum" title="Ruth Teitelbaum">Ruth Lichterman</a>, <a href="/wiki/Betty_Holberton" title="Betty Holberton">Betty Snyder</a>, <a href="/wiki/Frances_Spence" title="Frances Spence">Frances Bilas</a>, and <a href="/wiki/Kathleen_Antonelli" title="Kathleen Antonelli">Kay McNulty</a> led said efforts.<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> <div class="mw-heading mw-heading3"><h3 id="Foundations_and_early_applications">Foundations and early applications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computer_algebra&action=edit&section=12" title="Edit section: Foundations and early applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1960, <a href="/wiki/John_McCarthy_(computer_scientist)" title="John McCarthy (computer scientist)">John McCarthy</a> explored an extension of <a href="/wiki/Primitive_recursive_function" title="Primitive recursive function">primitive recursive functions</a> for computing symbolic expressions through the <a href="/wiki/Lisp_(programming_language)" title="Lisp (programming language)">Lisp</a> programming language while at the <a href="/wiki/Massachusetts_Institute_of_Technology" title="Massachusetts Institute of Technology">Massachusetts Institute of Technology</a>.<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 his series on "Recursive functions of symbolic expressions and their computation by machine" remained incomplete,<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> McCarthy and his contributions to artificial intelligence programming and computer algebra via Lisp helped establish <a href="/wiki/MIT_Computer_Science_and_Artificial_Intelligence_Laboratory" title="MIT Computer Science and Artificial Intelligence Laboratory">Project MAC</a> at the Massachusetts Institute of Technology and the organization that later became the <a href="/wiki/Stanford_University_centers_and_institutes" title="Stanford University centers and institutes">Stanford AI Laboratory</a> (SAIL) at <a href="/wiki/Stanford_University" title="Stanford University">Stanford University</a>, whose competition facilitated significant development in computer algebra throughout the late 20th century. </p><p>Early efforts at symbolic computation, in the 1960s and 1970s, faced challenges surrounding the inefficiency of long-known algorithms when ported to computer algebra systems.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> Predecessors to Project MAC, such as <a href="/wiki/ALTRAN" title="ALTRAN">ALTRAN</a>, sought to overcome algorithmic limitations through advancements in hardware and interpreters, while later efforts turned towards software optimization.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Historic_problems">Historic problems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computer_algebra&action=edit&section=13" title="Edit section: Historic problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A large part of the work of researchers in the field consisted of revisiting classical <a href="/wiki/Algebra" title="Algebra">algebra</a> to increase its <a href="/wiki/Computable_function" title="Computable function">effectiveness</a> while developing <a href="/wiki/Algorithmic_efficiency" title="Algorithmic efficiency">efficient algorithms</a> for use in computer algebra. An example of this type of work is the computation of <a href="/wiki/Polynomial_greatest_common_divisor" title="Polynomial greatest common divisor">polynomial greatest common divisors</a>, a task required to simplify fractions and an essential component of computer algebra. Classical algorithms for this computation, such as <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclid's algorithm</a>, proved inefficient over infinite fields; algorithms from <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> faced similar struggles.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Thus, researchers turned to discovering methods of reducing polynomials (such as those over a <a href="/wiki/Ring_of_integers" title="Ring of integers">ring of integers</a> or a <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">unique factorization domain</a>) to a variant efficiently computable via a Euclidean algorithm. </p> <div class="mw-heading mw-heading2"><h2 id="Algorithms_used_in_computer_algebra">Algorithms used in computer algebra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computer_algebra&action=edit&section=14" title="Edit section: Algorithms used in computer algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/List_of_algorithms#Computer_algebra" title="List of algorithms">List of algorithms § Computer algebra</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=List_of_algorithms&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <ul><li><a href="/wiki/Buchberger%27s_algorithm" title="Buchberger's algorithm">Buchberger's algorithm</a>: finds a <a href="/wiki/Gr%C3%B6bner_basis" title="Gröbner basis">Gröbner basis</a></li> <li><a href="/wiki/Cantor%E2%80%93Zassenhaus_algorithm" title="Cantor–Zassenhaus algorithm">Cantor–Zassenhaus algorithm</a>: factor polynomials over finite fields</li> <li><a href="/wiki/Faug%C3%A8re_F4_algorithm" class="mw-redirect" title="Faugère F4 algorithm">Faugère F4 algorithm</a>: finds a Gröbner basis (also mentions the F5 algorithm)</li> <li><a href="/wiki/Gosper%27s_algorithm" title="Gosper's algorithm">Gosper's algorithm</a>: find sums of hypergeometric terms that are themselves hypergeometric terms</li> <li><a href="/wiki/Knuth%E2%80%93Bendix_completion_algorithm" title="Knuth–Bendix completion algorithm">Knuth–Bendix completion algorithm</a>: for <a href="/wiki/Rewriting" title="Rewriting">rewriting</a> rule systems</li> <li><a href="/wiki/Multivariate_division_algorithm" class="mw-redirect" title="Multivariate division algorithm">Multivariate division algorithm</a>: for <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> in several indeterminates</li> <li><a href="/wiki/Pollard%27s_kangaroo_algorithm" title="Pollard's kangaroo algorithm">Pollard's kangaroo algorithm</a> (also known as Pollard's lambda algorithm ): an algorithm for solving the discrete logarithm problem</li> <li><a href="/wiki/Polynomial_long_division" title="Polynomial long division">Polynomial long division</a>: an algorithm for dividing a polynomial by another polynomial of the same or lower degree</li> <li><a href="/wiki/Risch_algorithm" title="Risch algorithm">Risch algorithm</a>: an algorithm for the calculus operation of indefinite integration (i.e. finding <a href="/wiki/Antiderivatives" class="mw-redirect" title="Antiderivatives">antiderivatives</a>)</li></ul></div></div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computer_algebra&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Automated_theorem_prover" class="mw-redirect" title="Automated theorem prover">Automated theorem prover</a></li> <li><a href="/wiki/Computer-assisted_proof" title="Computer-assisted proof">Computer-assisted proof</a></li> <li><a href="/wiki/Computational_algebraic_geometry" class="mw-redirect" title="Computational algebraic geometry">Computational algebraic geometry</a></li> <li><a href="/wiki/Computer_algebra_system" title="Computer algebra system">Computer algebra system</a></li> <li><a href="/wiki/Differential_analyser" title="Differential analyser">Differential analyser</a></li> <li><a href="/wiki/Proof_checker" class="mw-redirect" title="Proof checker">Proof checker</a></li> <li><a href="/wiki/Model_checker" class="mw-redirect" title="Model checker">Model checker</a></li> <li><a href="/wiki/Symbolic-numeric_computation" title="Symbolic-numeric computation">Symbolic-numeric computation</a></li> <li><a href="/wiki/Symbolic_simulation" title="Symbolic simulation">Symbolic simulation</a></li> <li><a href="/wiki/Symbolic_artificial_intelligence" title="Symbolic artificial intelligence">Symbolic artificial intelligence</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computer_algebra&action=edit&section=16" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em 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.cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.sigsam.org/cca/">"ACM Association in computer algebra"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=ACM+Association+in+computer+algebra&rft_id=https%3A%2F%2Fwww.sigsam.org%2Fcca%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWatt2006" class="citation conference cs1">Watt, Stephen M. (2006). <a rel="nofollow" class="external text" href="http://www.csd.uwo.ca/~watt/pub/reprints/2006-tc-sympoly.pdf"><i>Making Computer Algebra More Symbolic (Invited)</i></a> <span class="cs1-format">(PDF)</span>. Transgressive Computing 2006: A conference in honor of Jean Della Dora, (TC 2006). pp. 43–49. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9788468983813" title="Special:BookSources/9788468983813"><bdi>9788468983813</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/496720771">496720771</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Making+Computer+Algebra+More+Symbolic+%28Invited%29&rft.pages=43-49&rft.date=2006&rft_id=info%3Aoclcnum%2F496720771&rft.isbn=9788468983813&rft.aulast=Watt&rft.aufirst=Stephen+M.&rft_id=http%3A%2F%2Fwww.csd.uwo.ca%2F~watt%2Fpub%2Freprints%2F2006-tc-sympoly.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.sigsam.org">SIGSAM official site</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130808052201/http://www.sigsam.org/conferences/index.phtml">"SIGSAM list of conferences"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.sigsam.org/conferences/index.phtml">the original</a> on 2013-08-08<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-11-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=SIGSAM+list+of+conferences&rft_id=http%3A%2F%2Fwww.sigsam.org%2Fconferences%2Findex.phtml&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohen2003" class="citation book cs1">Cohen, Joel S. (2003). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/computeralgebras00cohe_792"><i>Computer Algebra and Symbolic Computation: Mathematical Methods</i></a></span>. AK Peters. p. <a rel="nofollow" class="external text" href="https://archive.org/details/computeralgebras00cohe_792/page/n33">14</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56881-159-8" title="Special:BookSources/978-1-56881-159-8"><bdi>978-1-56881-159-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computer+Algebra+and+Symbolic+Computation%3A+Mathematical+Methods&rft.pages=14&rft.pub=AK+Peters&rft.date=2003&rft.isbn=978-1-56881-159-8&rft.aulast=Cohen&rft.aufirst=Joel+S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcomputeralgebras00cohe_792&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.sigsam.org/journals.phtml">SIGSAM list of journals</a></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://homepages.math.uic.edu/~jan/mcs320/mcs320notes/lec12.html">"Lecture 12: Rational Functions and Conversions — Introduction to Symbolic Computation 1.7.6 documentation"</a>. <i>homepages.math.uic.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-03-31</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=homepages.math.uic.edu&rft.atitle=Lecture+12%3A+Rational+Functions+and+Conversions+%E2%80%94+Introduction+to+Symbolic+Computation+1.7.6+documentation&rft_id=https%3A%2F%2Fhomepages.math.uic.edu%2F~jan%2Fmcs320%2Fmcs320notes%2Flec12.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeutPetitotDridi2009" class="citation journal cs1">Neut, Sylvain; Petitot, Michel; Dridi, Raouf (2009-03-01). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/S0747717108001132">"Élie Cartan's geometrical vision or how to avoid expression swell"</a>. <i>Journal of Symbolic Computation</i>. Polynomial System Solving in honor of Daniel Lazard. <b>44</b> (3): 261–270. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jsc.2007.04.006">10.1016/j.jsc.2007.04.006</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0747-7171">0747-7171</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Computation&rft.atitle=%C3%89lie+Cartan%27s+geometrical+vision+or+how+to+avoid+expression+swell&rft.volume=44&rft.issue=3&rft.pages=261-270&rft.date=2009-03-01&rft_id=info%3Adoi%2F10.1016%2Fj.jsc.2007.04.006&rft.issn=0747-7171&rft.aulast=Neut&rft.aufirst=Sylvain&rft.au=Petitot%2C+Michel&rft.au=Dridi%2C+Raouf&rft_id=https%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fpii%2FS0747717108001132&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Richard Liska <a rel="nofollow" class="external text" href="http://kfe.fjfi.cvut.cz/~liska/ca/node53.html#:~:text=is%20a%20common%20phenomenon%20of,dramatically%20as%20the%20calculation%20progresses.">Expression swell</a>, from "Peculiarities of programming in computer algebra systems"</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://library.wolfram.com/infocenter/Conferences/7518/Macalester_talk.txt">"The Mathematica Kernel: Issues in the Design and Implementation"</a>. October 2006. Retrieved 2023-11-29.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.maplesoft.com/support/help/AddOns/view.aspx?path=GMP">"The GNU Multiple Precision (GMP) Library"</a>. <a href="/wiki/Maplesoft" class="mw-redirect" title="Maplesoft">Maplesoft</a>. Retrieved 2023-11-29.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCassidy1985" class="citation thesis cs1">Cassidy, Kevin G. (Dec 1985). <a class="external text" href="https://commons.wikimedia.org/wiki/File:The_feasibility_of_automatic_storage_reclamation_with_concurrent_program_execution_in_a_LISP_environment._(IA_feasibilityofaut00cass).pdf"><i>The Feasibility of Automatic Storage Reclamation with Concurrent Program Execution in a LISP Environment</i></a> <span class="cs1-format">(PDF)</span> (Master's thesis). Naval Postgraduate School, Monterey/CA. p. 15. ADA165184.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=The+Feasibility+of+Automatic+Storage+Reclamation+with+Concurrent+Program+Execution+in+a+LISP+Environment&rft.inst=Naval+Postgraduate+School%2C+Monterey%2FCA&rft.date=1985-12&rft.aulast=Cassidy&rft.aufirst=Kevin+G.&rft_id=https%3A%2F%2Fcommons.wikimedia.org%2Fwiki%2FFile%3AThe_feasibility_of_automatic_storage_reclamation_with_concurrent_program_execution_in_a_LISP_environment._%28IA_feasibilityofaut00cass%29.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://people.eecs.berkeley.edu/~fateman/macsyma/docs/refman16.pdf"><i>Macsyma Mathematics and System Reference Manual</i></a> <span class="cs1-format">(PDF)</span>. <a href="/wiki/Macsyma" title="Macsyma">Macsyma</a>. 1996. p. 419.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Macsyma+Mathematics+and+System+Reference+Manual&rft.pages=419&rft.pub=Macsyma&rft.date=1996&rft_id=https%3A%2F%2Fpeople.eecs.berkeley.edu%2F~fateman%2Fmacsyma%2Fdocs%2Frefman16.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBuchbergerLoos1983" class="citation book cs1">Buchberger, Bruno; Loos, Rüdiger (1983). <a rel="nofollow" class="external text" href="https://www.risc.jku.at/people/buchberg/papers/1982-00-00-B.pdf">"Algebraic simplification"</a> <span class="cs1-format">(PDF)</span>. In Buchberger, Bruno; Collins, George Edwin; Loos, Rüdiger; Albrecht, Rudolf (eds.). <i>Computer Algebra: Symbolic and Algebraic Computation</i>. Computing Supplementa. Vol. 4. pp. 11–43. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-7091-7551-4_2">10.1007/978-3-7091-7551-4_2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-211-81776-6" title="Special:BookSources/978-3-211-81776-6"><bdi>978-3-211-81776-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Algebraic+simplification&rft.btitle=Computer+Algebra%3A+Symbolic+and+Algebraic+Computation&rft.series=Computing+Supplementa&rft.pages=11-43&rft.date=1983&rft_id=info%3Adoi%2F10.1007%2F978-3-7091-7551-4_2&rft.isbn=978-3-211-81776-6&rft.aulast=Buchberger&rft.aufirst=Bruno&rft.au=Loos%2C+R%C3%BCdiger&rft_id=https%3A%2F%2Fwww.risc.jku.at%2Fpeople%2Fbuchberg%2Fpapers%2F1982-00-00-B.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavenportSiretTournier1988" class="citation book cs1">Davenport, J. H.; Siret, Y.; Tournier, É. (1988). <i>Computer Algebra: Systems and Algorithms for Algebraic Computation</i>. Academic. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-204230-1" title="Special:BookSources/0-12-204230-1"><bdi>0-12-204230-1</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/802584470">802584470</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computer+Algebra%3A+Systems+and+Algorithms+for+Algebraic+Computation&rft.pub=Academic&rft.date=1988&rft_id=info%3Aoclcnum%2F802584470&rft.isbn=0-12-204230-1&rft.aulast=Davenport&rft.aufirst=J.+H.&rft.au=Siret%2C+Y.&rft.au=Tournier%2C+%C3%89.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.seas.upenn.edu/about-seas/eniac/operation.php">"ENIAC in Action: What it Was and How it Worked"</a>. <i>ENIAC: Celebrating Penn Engineering History</i>. University of Pennsylvania. Retrieved December 3, 2023.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLight1999" class="citation journal cs1">Light, Jennifer S. (1999). <a rel="nofollow" class="external text" href="https://muse.jhu.edu/article/33396">"When Computers Were Women"</a>. <i>Technology and Culture</i>. <b>40</b> (3): 455–483. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1353%2Ftech.1999.0128">10.1353/tech.1999.0128</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1097-3729">1097-3729</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Technology+and+Culture&rft.atitle=When+Computers+Were+Women&rft.volume=40&rft.issue=3&rft.pages=455-483&rft.date=1999&rft_id=info%3Adoi%2F10.1353%2Ftech.1999.0128&rft.issn=1097-3729&rft.aulast=Light&rft.aufirst=Jennifer+S.&rft_id=https%3A%2F%2Fmuse.jhu.edu%2Farticle%2F33396&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcCarthy1960" class="citation journal cs1">McCarthy, John (1960-04-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F367177.367199">"Recursive functions of symbolic expressions and their computation by machine, Part I"</a>. <i>Communications of the ACM</i>. <b>3</b> (4): 184–195. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F367177.367199">10.1145/367177.367199</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0001-0782">0001-0782</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+of+the+ACM&rft.atitle=Recursive+functions+of+symbolic+expressions+and+their+computation+by+machine%2C+Part+I&rft.volume=3&rft.issue=4&rft.pages=184-195&rft.date=1960-04-01&rft_id=info%3Adoi%2F10.1145%2F367177.367199&rft.issn=0001-0782&rft.aulast=McCarthy&rft.aufirst=John&rft_id=https%3A%2F%2Fdoi.org%2F10.1145%252F367177.367199&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWexelblat1981" class="citation book cs1">Wexelblat, Richard L. (1981). <i>History of programming languages</i>. ACM monograph series. History of programming languages conference, Association for computing machinery. New York London Toronto: Academic press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-745040-7" title="Special:BookSources/978-0-12-745040-7"><bdi>978-0-12-745040-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+programming+languages&rft.place=New+York+London+Toronto&rft.series=ACM+monograph+series&rft.pub=Academic+press&rft.date=1981&rft.isbn=978-0-12-745040-7&rft.aulast=Wexelblat&rft.aufirst=Richard+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1"><a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/S0747717185800250">"Symbolic Computation (An Editorial)"</a>. <i>Journal of Symbolic Computation</i>. <b>1</b> (1): 1–6. 1985-03-01. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0747-7171%2885%2980025-0">10.1016/S0747-7171(85)80025-0</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0747-7171">0747-7171</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Computation&rft.atitle=Symbolic+Computation+%28An+Editorial%29&rft.volume=1&rft.issue=1&rft.pages=1-6&rft.date=1985-03-01&rft_id=info%3Adoi%2F10.1016%2FS0747-7171%2885%2980025-0&rft.issn=0747-7171&rft_id=https%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fpii%2FS0747717185800250&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeldman1975" class="citation journal cs1">Feldman, Stuart I. (1975-11-01). <a rel="nofollow" class="external text" href="https://dl.acm.org/doi/10.1145/1088322.1088325">"A brief description of Altran"</a>. <i>ACM SIGSAM Bulletin</i>. <b>9</b> (4): 12–20. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F1088322.1088325">10.1145/1088322.1088325</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0163-5824">0163-5824</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+SIGSAM+Bulletin&rft.atitle=A+brief+description+of+Altran&rft.volume=9&rft.issue=4&rft.pages=12-20&rft.date=1975-11-01&rft_id=info%3Adoi%2F10.1145%2F1088322.1088325&rft.issn=0163-5824&rft.aulast=Feldman&rft.aufirst=Stuart+I.&rft_id=https%3A%2F%2Fdl.acm.org%2Fdoi%2F10.1145%2F1088322.1088325&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaltofen1983" class="citation cs2">Kaltofen, E. (1983), Buchberger, Bruno; Collins, George Edwin; Loos, Rüdiger; Albrecht, Rudolf (eds.), <a rel="nofollow" class="external text" href="http://link.springer.com/10.1007/978-3-7091-7551-4_8">"Factorization of Polynomials"</a>, <i>Computer Algebra</i>, Computing Supplementa, vol. 4, Vienna: Springer Vienna, pp. 95–113, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-7091-7551-4_8">10.1007/978-3-7091-7551-4_8</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-211-81776-6" title="Special:BookSources/978-3-211-81776-6"><bdi>978-3-211-81776-6</bdi></a><span class="reference-accessdate">, retrieved <span class="nowrap">2023-11-29</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computer+Algebra&rft.atitle=Factorization+of+Polynomials&rft.volume=4&rft.pages=95-113&rft.date=1983&rft_id=info%3Adoi%2F10.1007%2F978-3-7091-7551-4_8&rft.isbn=978-3-211-81776-6&rft.aulast=Kaltofen&rft.aufirst=E.&rft_id=http%3A%2F%2Flink.springer.com%2F10.1007%2F978-3-7091-7551-4_8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computer_algebra&action=edit&section=17" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a detailed definition of the subject: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBuchberger1985" class="citation journal cs1"><a href="/wiki/Bruno_Buchberger" title="Bruno Buchberger">Buchberger, Bruno</a> (1985). <a rel="nofollow" class="external text" href="http://www3.risc.jku.at/publications/download/risc_2749/1985-03-00-C.pdf">"Symbolic Computation (An Editorial)"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Journal_of_Symbolic_Computation" title="Journal of Symbolic Computation">Journal of Symbolic Computation</a></i>. <b>1</b> (1): 1–6. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0747-7171%2885%2980025-0">10.1016/S0747-7171(85)80025-0</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Computation&rft.atitle=Symbolic+Computation+%28An+Editorial%29&rft.volume=1&rft.issue=1&rft.pages=1-6&rft.date=1985&rft_id=info%3Adoi%2F10.1016%2FS0747-7171%2885%2980025-0&rft.aulast=Buchberger&rft.aufirst=Bruno&rft_id=http%3A%2F%2Fwww3.risc.jku.at%2Fpublications%2Fdownload%2Frisc_2749%2F1985-03-00-C.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></li></ul> <p>For textbooks devoted to the subject: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavenportSiretTournier1988" class="citation book cs1"><a href="/wiki/James_H._Davenport" title="James H. Davenport">Davenport, James H.</a>; Siret, Yvon; Tournier, Èvelyne (1988). <i>Computer Algebra: Systems and Algorithms for Algebraic Computation</i>. Translated from the French by A. Davenport and J. H. Davenport. Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-204230-0" title="Special:BookSources/978-0-12-204230-0"><bdi>978-0-12-204230-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computer+Algebra%3A+Systems+and+Algorithms+for+Algebraic+Computation&rft.pub=Academic+Press&rft.date=1988&rft.isbn=978-0-12-204230-0&rft.aulast=Davenport&rft.aufirst=James+H.&rft.au=Siret%2C+Yvon&rft.au=Tournier%2C+%C3%88velyne&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_zur_GathenGerhard2003" class="citation book cs1">von zur Gathen, Joachim; Gerhard, Jürgen (2003). <i>Modern computer algebra</i> (2nd ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-82646-2" title="Special:BookSources/0-521-82646-2"><bdi>0-521-82646-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+computer+algebra&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=0-521-82646-2&rft.aulast=von+zur+Gathen&rft.aufirst=Joachim&rft.au=Gerhard%2C+J%C3%BCrgen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeddesCzaporLabahn1992" class="citation book cs1">Geddes, K. O.; Czapor, S. R.; Labahn, G. (1992). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/algorithmsforcom0000gedd"><i>Algorithms for Computer Algebra</i></a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1992afca.book.....G">1992afca.book.....G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fb102438">10.1007/b102438</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7923-9259-0" title="Special:BookSources/978-0-7923-9259-0"><bdi>978-0-7923-9259-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algorithms+for+Computer+Algebra&rft.date=1992&rft_id=info%3Adoi%2F10.1007%2Fb102438&rft_id=info%3Abibcode%2F1992afca.book.....G&rft.isbn=978-0-7923-9259-0&rft.aulast=Geddes&rft.aufirst=K.+O.&rft.au=Czapor%2C+S.+R.&rft.au=Labahn%2C+G.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Falgorithmsforcom0000gedd&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputer+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBuchbergerCollinsLoosAlbrecht1983" class="citation book cs1">Buchberger, Bruno; Collins, George Edwin; Loos, Rüdiger; Albrecht, Rudolf, eds. (1983). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/computeralgebras0000unse"><i>Computer Algebra: Symbolic and Algebraic Computation</i></a></span>. Computing Supplementa. Vol. 4. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-7091-7551-4">10.1007/978-3-7091-7551-4</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-211-81776-6" title="Special:BookSources/978-3-211-81776-6"><bdi>978-3-211-81776-6</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5221892">5221892</a>.</cite><span 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title="CoCoA">CoCoA</a></li> <li><a href="/wiki/Fermat_(computer_algebra_system)" title="Fermat (computer algebra system)">Fermat</a></li> <li><a href="/wiki/FriCAS" title="FriCAS">FriCAS</a></li> <li><a href="/wiki/FORM_(symbolic_manipulation_system)" title="FORM (symbolic manipulation system)">FORM</a></li> <li><a href="/wiki/GAP_(computer_algebra_system)" title="GAP (computer algebra system)">GAP</a></li> <li><a href="/wiki/GiNaC" title="GiNaC">GiNaC</a></li> <li><a href="/wiki/Macaulay2" title="Macaulay2">Macaulay2</a></li> <li><a href="/wiki/Maxima_(software)" title="Maxima (software)">Maxima</a></li> <li><a href="/wiki/Normaliz" title="Normaliz">Normaliz</a></li> <li><a href="/wiki/PARI/GP" title="PARI/GP">PARI/GP</a></li> <li><a href="/wiki/Reduce_(computer_algebra_system)" title="Reduce (computer algebra system)">Reduce</a></li> <li><a href="/wiki/SageMath" title="SageMath">SageMath</a></li> <li><a href="/wiki/Singular_(software)" title="Singular (software)">Singular</a></li> 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title="MuPAD">muPAD</a> (<a href="/wiki/MATLAB" title="MATLAB">MATLAB</a> symbolic math toolbox)</li> <li><a href="/wiki/SMath_Studio" title="SMath Studio">SMath Studio</a></li> <li><a href="/wiki/TI_InterActive!" title="TI InterActive!">TI InterActive!</a></li> <li><a href="/wiki/Engineering_Equation_Solver" title="Engineering Equation Solver">Engineering Equation Solver</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="/wiki/Cambridge_Algebra_System" title="Cambridge Algebra System">CAMAL</a></li> <li><a href="/wiki/Derive_(computer_algebra_system)" title="Derive (computer algebra system)">Derive</a></li> <li><a href="/wiki/Erable" title="Erable">Erable</a></li> <li><a href="/wiki/LiveMath" title="LiveMath">LiveMath</a></li> <li><a href="/wiki/Macsyma" title="Macsyma">Macsyma</a></li> <li><a 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article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span><b><a href="/wiki/List_of_computer_algebra_systems" title="List of computer algebra systems">List</a></b></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Computer_science" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" 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science</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>Note: This template roughly follows the 2012 <a href="/wiki/ACM_Computing_Classification_System" title="ACM Computing Classification System">ACM Computing Classification System</a>.</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_hardware" title="Computer hardware">Hardware</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="/wiki/Printed_circuit_board" title="Printed circuit board">Printed circuit board</a></li> <li><a href="/wiki/Peripheral" title="Peripheral">Peripheral</a></li> <li><a href="/wiki/Integrated_circuit" title="Integrated circuit">Integrated circuit</a></li> <li><a href="/wiki/Very_Large_Scale_Integration" class="mw-redirect" title="Very Large Scale Integration">Very Large Scale Integration</a></li> <li><a href="/wiki/System_on_a_chip" title="System on a 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class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algorithm_design" class="mw-redirect" title="Algorithm design">Algorithm design</a></li> <li><a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">Analysis of algorithms</a></li> <li><a href="/wiki/Algorithmic_efficiency" title="Algorithmic efficiency">Algorithmic efficiency</a></li> <li><a href="/wiki/Randomized_algorithm" title="Randomized algorithm">Randomized algorithm</a></li> <li><a href="/wiki/Computational_geometry" title="Computational geometry">Computational geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematics of <a href="/wiki/Computing" title="Computing">computing</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="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete mathematics</a></li> <li><a href="/wiki/Probability" title="Probability">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Mathematical_software" title="Mathematical software">Mathematical software</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li> <li><a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">Theoretical computer science</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Information_system" title="Information system">Information systems</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="/wiki/Database" title="Database">Database management system</a></li> <li><a href="/wiki/Computer_data_storage" title="Computer data storage">Information storage systems</a></li> <li><a href="/wiki/Enterprise_information_system" title="Enterprise information system">Enterprise information system</a></li> <li><a href="/wiki/Social_software" title="Social software">Social information systems</a></li> <li><a href="/wiki/Geographic_information_system" title="Geographic information system">Geographic information system</a></li> <li><a href="/wiki/Decision_support_system" title="Decision support system">Decision support system</a></li> <li><a href="/wiki/Process_control" class="mw-redirect" title="Process control">Process control system</a></li> <li><a href="/wiki/Multimedia_database" title="Multimedia database">Multimedia information system</a></li> <li><a href="/wiki/Data_mining" title="Data mining">Data mining</a></li> <li><a href="/wiki/Digital_library" title="Digital library">Digital library</a></li> <li><a href="/wiki/Computing_platform" title="Computing platform">Computing platform</a></li> <li><a href="/wiki/Digital_marketing" title="Digital marketing">Digital marketing</a></li> <li><a href="/wiki/World_Wide_Web" title="World Wide Web">World Wide Web</a></li> <li><a href="/wiki/Information_retrieval" title="Information retrieval">Information retrieval</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_security" title="Computer security">Security</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="/wiki/Cryptography" title="Cryptography">Cryptography</a></li> <li><a href="/wiki/Formal_methods" title="Formal methods">Formal methods</a></li> <li><a href="/wiki/Security_hacker" title="Security hacker">Security hacker</a></li> <li><a href="/wiki/Security_service_(telecommunication)" title="Security service (telecommunication)">Security services</a></li> <li><a href="/wiki/Intrusion_detection_system" title="Intrusion detection system">Intrusion detection system</a></li> <li><a href="/wiki/Hardware_security" title="Hardware security">Hardware security</a></li> <li><a href="/wiki/Network_security" title="Network security">Network security</a></li> <li><a href="/wiki/Information_security" title="Information security">Information security</a></li> <li><a href="/wiki/Application_security" title="Application security">Application security</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Human%E2%80%93computer_interaction" title="Human–computer interaction">Human–computer interaction</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="/wiki/Interaction_design" title="Interaction design">Interaction design</a></li> <li><a href="/wiki/Social_computing" title="Social computing">Social computing</a></li> <li><a href="/wiki/Ubiquitous_computing" title="Ubiquitous computing">Ubiquitous computing</a></li> <li><a href="/wiki/Visualization_(graphics)" title="Visualization (graphics)">Visualization</a></li> <li><a href="/wiki/Computer_accessibility" title="Computer accessibility">Accessibility</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Concurrency_(computer_science)" title="Concurrency (computer science)">Concurrency</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="/wiki/Concurrent_computing" title="Concurrent computing">Concurrent computing</a></li> <li><a href="/wiki/Parallel_computing" title="Parallel computing">Parallel computing</a></li> <li><a href="/wiki/Distributed_computing" title="Distributed computing">Distributed computing</a></li> <li><a href="/wiki/Multithreading_(computer_architecture)" title="Multithreading (computer architecture)">Multithreading</a></li> <li><a href="/wiki/Multiprocessing" title="Multiprocessing">Multiprocessing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Artificial_intelligence" title="Artificial intelligence">Artificial intelligence</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="/wiki/Natural_language_processing" title="Natural language processing">Natural language processing</a></li> <li><a href="/wiki/Knowledge_representation_and_reasoning" title="Knowledge representation and reasoning">Knowledge representation and reasoning</a></li> <li><a href="/wiki/Computer_vision" title="Computer vision">Computer vision</a></li> <li><a href="/wiki/Automated_planning_and_scheduling" title="Automated planning and scheduling">Automated planning and scheduling</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Search methodology</a></li> <li><a href="/wiki/Control_theory" title="Control theory">Control method</a></li> <li><a href="/wiki/Philosophy_of_artificial_intelligence" title="Philosophy of artificial intelligence">Philosophy of artificial intelligence</a></li> <li><a href="/wiki/Distributed_artificial_intelligence" title="Distributed artificial intelligence">Distributed artificial intelligence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</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="/wiki/Supervised_learning" title="Supervised learning">Supervised learning</a></li> <li><a href="/wiki/Unsupervised_learning" title="Unsupervised learning">Unsupervised learning</a></li> <li><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></li> <li><a href="/wiki/Multi-task_learning" title="Multi-task learning">Multi-task learning</a></li> <li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross-validation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_graphics" title="Computer graphics">Graphics</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="/wiki/Computer_animation" title="Computer animation">Animation</a></li> <li><a href="/wiki/Rendering_(computer_graphics)" title="Rendering (computer graphics)">Rendering</a></li> <li><a href="/wiki/Photograph_manipulation" title="Photograph manipulation">Photograph manipulation</a></li> <li><a href="/wiki/Graphics_processing_unit" title="Graphics processing unit">Graphics processing unit</a></li> <li><a href="/wiki/Mixed_reality" title="Mixed reality">Mixed reality</a></li> <li><a href="/wiki/Virtual_reality" title="Virtual reality">Virtual reality</a></li> <li><a href="/wiki/Image_compression" title="Image compression">Image compression</a></li> <li><a href="/wiki/Solid_modeling" title="Solid modeling">Solid modeling</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applied computing</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="/wiki/Quantum_Computing" class="mw-redirect" title="Quantum Computing">Quantum Computing</a></li> <li><a href="/wiki/E-commerce" title="E-commerce">E-commerce</a></li> <li><a href="/wiki/Enterprise_software" title="Enterprise software">Enterprise software</a></li> <li><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational mathematics</a></li> <li><a href="/wiki/Computational_physics" title="Computational physics">Computational physics</a></li> <li><a href="/wiki/Computational_chemistry" title="Computational chemistry">Computational chemistry</a></li> <li><a href="/wiki/Computational_biology" title="Computational biology">Computational biology</a></li> <li><a href="/wiki/Computational_social_science" title="Computational social science">Computational social science</a></li> <li><a href="/wiki/Computational_engineering" title="Computational engineering">Computational engineering</a></li> <li><a href="/wiki/Template:Differentiable_computing" title="Template:Differentiable computing">Differentiable computing</a></li> <li><a href="/wiki/Health_informatics" title="Health informatics">Computational healthcare</a></li> <li><a href="/wiki/Digital_art" title="Digital art">Digital art</a></li> <li><a href="/wiki/Electronic_publishing" title="Electronic publishing">Electronic publishing</a></li> <li><a href="/wiki/Cyberwarfare" title="Cyberwarfare">Cyberwarfare</a></li> <li><a href="/wiki/Electronic_voting" title="Electronic voting">Electronic voting</a></li> <li><a href="/wiki/Video_game" title="Video game">Video games</a></li> <li><a href="/wiki/Word_processor" title="Word processor">Word processing</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li> <li><a href="/wiki/Educational_technology" title="Educational technology">Educational technology</a></li> <li><a href="/wiki/Document_management_system" title="Document management system">Document management</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"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Computer_science" title="Category:Computer science">Category</a></li> <li><span class="noviewer" typeof="mw:File"><span title="Outline"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Global_thinking.svg/10px-Global_thinking.svg.png" decoding="async" width="10" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Global_thinking.svg/15px-Global_thinking.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/41/Global_thinking.svg/21px-Global_thinking.svg.png 2x" data-file-width="130" data-file-height="200" /></span></span> <a href="/wiki/Outline_of_computer_science" title="Outline of computer science">Outline</a></li> <li><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e0/Symbol_question.svg/16px-Symbol_question.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e0/Symbol_question.svg/23px-Symbol_question.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e0/Symbol_question.svg/31px-Symbol_question.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Template:Glossaries_of_computers" title="Template:Glossaries of computers">Glossaries</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Major_mathematics_areas" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Areas_of_mathematics" title="Template:Areas of mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Areas_of_mathematics" title="Template talk:Areas of mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Areas_of_mathematics" title="Special:EditPage/Template:Areas of mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_mathematics_areas" style="font-size:114%;margin:0 4em">Major <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> areas</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_mathematics" title="History of mathematics">History</a> <ul><li><a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">Timeline</a></li> <li><a href="/wiki/Future_of_mathematics" title="Future of mathematics">Future</a></li></ul></li> <li><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Lists</a></li> <li><a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations</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="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algebra" title="Algebra">Algebra</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="/wiki/Abstract_algebra" title="Abstract algebra">Abstract</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative</a></li> <li><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary</a></li> <li><a href="/wiki/Group_theory" title="Group theory">Group theory</a></li> <li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear</a></li> <li><a href="/wiki/Universal_algebra" title="Universal algebra">Universal</a></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</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="/wiki/Calculus" title="Calculus">Calculus</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Complex_analysis" title="Complex analysis">Complex analysis</a></li> <li><a href="/wiki/Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a></li> <li><a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete</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="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Order_theory" title="Order theory">Order theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Geometry" title="Geometry">Geometry</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="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic</a></li> <li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Number_theory" title="Number theory">Number theory</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="/wiki/Arithmetic" title="Arithmetic">Arithmetic</a></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</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="/wiki/General_topology" title="General topology">General</a></li> <li><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic</a></li> <li><a href="/wiki/Differential_topology" title="Differential topology">Differential</a></li> <li><a href="/wiki/Geometric_topology" title="Geometric topology">Geometric</a></li> <li><a href="/wiki/Homotopy_theory" title="Homotopy theory">Homotopy theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied</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="/wiki/Engineering_mathematics" title="Engineering mathematics">Engineering mathematics</a></li> <li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Mathematical biology</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Mathematical chemistry</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li> <li><a href="/wiki/Mathematical_psychology" title="Mathematical psychology">Mathematical psychology</a></li> <li><a href="/wiki/Mathematical_sociology" title="Mathematical sociology">Mathematical sociology</a></li> <li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Systems_science" title="Systems science">Systems science</a> <ul><li><a href="/wiki/Control_theory" title="Control theory">Control theory</a></li> <li><a href="/wiki/Game_theory" title="Game theory">Game theory</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational</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="/wiki/Computer_science" title="Computer science">Computer science</a></li> <li><a href="/wiki/Theory_of_computation" title="Theory of computation">Theory of computation</a></li> <li><a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a></li> <li><a class="mw-selflink selflink">Computer algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Related 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