4417:
4429:
3821:
2814:
3831:
2826:
460:
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1498:, 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.
36:
537:
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
1142:
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
495:
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
491:
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
926:
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
1457:
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
81:
1493:
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
711:
1572:. Though his series on "Recursive functions of symbolic expressions and their computation by machine" remained incomplete, McCarthy and his contributions to artificial intelligence programming and computer algebra via Lisp helped establish
380:
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
1523:
or programmers to reprogram it between calculations, manipulate its many physical modules (or panels), and feed its IBM card reader. Female mathematicians handled the majority of ENIAC programming human-guided computation:
538:
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.
246:
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
1587:
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. Predecessors to
Project MAC, such as
279:
for the computer science aspect of the subject and "computer algebra" for the mathematical aspect. In some languages the name of the field is not a direct translation of its
English name. Typically, it is called
1140:
1035:
2030:
1573:
1260:
1448:
1490:
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.
541:
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
745:
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
1988:
1346:
2570:
1301:
608:
903:. 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.
600:
3877:
1592:, sought to overcome algorithmic limitations through advancements in hardware and interpreters, while later efforts turned towards software optimization.
2864:
1577:
444:
408:
Therefore, the basic numbers used in computer algebra are the integers of the mathematicians, commonly represented by an unbounded signed sequence of
45:
2245:. ACM monograph series. History of programming languages conference, Association for computing machinery. New York London Toronto: Academic press.
3574:
3546:
2563:
1677:
3599:
2531:
2500:
2442:
2364:
2250:
2096:
1898:
1569:
723:. There are several classes of rewriting rules to be considered. The simplest are rules that always reduce the size of the expression, like
4240:
3450:
1616:, a task required to simplify fractions and an essential component of computer algebra. Classical algorithms for this computation, such as
4457:
930:
Some rewriting rules sometimes increase and sometimes decrease the size of the expressions to which they are applied. This is the case of
385:. To obviate this problem, various methods are used in the representation of the data, as well as in the algorithms that manipulate them.
215:
alluding to the complexity of the main applications that include, at least, a method to represent mathematical data in a computer, a user
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2620:
328:
4385:
3870:
3116:
1613:
1557:
1040:
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3121:
2818:
2556:
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2121:
1821:
67:
941:
52:
If the information is appropriate for the lead of the article, this information should also be included in the body of the article.
3921:
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2945:
2830:
2769:
2655:
2610:
1773:
1738:
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sorts the operands of sums and products into an order that places like terms in consecutive places, allowing easy detection. In
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2404:
1697:
1687:
347:
136:
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in order to combine or cancel them. Testing every pair of terms is costly with very long sums and products. To address this,
542:
492:
with "=" as an operator, a matrix may be represented as an expression with "matrix" as an operator and its rows as operands.
311:, but these terms, which also refer to non-computational manipulation, are no longer used in reference to computer algebra.
1856:
1486:
is such that two expressions in canonical form are semantically equal if and only if they are syntactically equal, while a
3863:
3665:
3642:
3372:
3362:
2625:
1469:
To test the equality of two expressions, instead of designing specific algorithms, it is usual to put expressions in some
228:
716:
A simpler expression than this is generally desired, and simplification is needed when working with general expressions.
3746:
3334:
3242:
3154:
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1561:
1624:
faced similar struggles. Thus, researchers turned to discovering methods of reducing polynomials (such as those over a
1176:
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3569:
3074:
2748:
1629:
1565:
468:
224:
3931:
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2600:
1516:
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4345:
4317:
3954:
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3751:
3726:
3589:
3237:
2850:
1650:
1373:
4390:
3675:
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3101:
2970:
2764:
1728:
1703:
1356:
1156:
1144:
484:
236:
172:
2000:
1584:, whose competition facilitated significant development in computer algebra throughout the late 20th century.
271:
using the latter name to refer to kinds of symbolic computation other than the computation with mathematical
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4169:
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3736:
3670:
3561:
3377:
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1743:
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1170:
appearing in an expression are simplified, they are usually considered as new indeterminates. For example,
1160:
432:
248:
207:
4445:
2396:
1926:"Lecture 12: Rational Functions and Conversions — Introduction to Symbolic Computation 1.7.6 documentation"
4373:
4270:
4250:
4245:
4174:
3899:
3801:
3632:
3513:
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3270:
3265:
1609:
1143:"factor", requires a non-trivial algorithm, which is thus a key function in computer algebra systems (see
935:
480:
431:
Programming an efficient implementation of the arithmetic operations is a hard task. Therefore, most free
324:
240:
196:
342:(International Symposium on Symbolic and Algebraic Computation), which is regularly sponsored by SIGSAM.
4400:
4330:
4207:
4131:
4070:
4055:
4050:
4027:
3909:
3771:
3741:
3731:
3627:
3541:
3417:
3357:
3324:
3314:
3204:
3169:
3159:
3096:
2965:
2940:
2935:
2900:
2032:
The
Feasibility of Automatic Storage Reclamation with Concurrent Program Execution in a LISP Environment
1748:
1520:
742:
394:
373:
188:
184:
4416:
2343:
Kaltofen, E. (1983), Buchberger, Bruno; Collins, George Edwin; Loos, Rüdiger; Albrecht, Rudolf (eds.),
1671:
405:
of a fixed bounded size. Neither of these is convenient for computer algebra, due to expression swell.
2266:
2049:
1950:
4380:
4260:
4255:
4179:
4080:
3531:
3503:
3475:
3470:
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3194:
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3141:
3091:
3086:
3003:
2949:
2690:
2587:
2480:
1816:. Transgressive Computing 2006: A conference in honor of Jean Della Dora, (TC 2006). pp. 43–49.
425:
216:
180:
84:
1925:
1700:(also known as Pollard's lambda algorithm ): an algorithm for solving the discrete logarithm problem
183:, they are generally considered as distinct fields because scientific computing is usually based on
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4126:
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4017:
4007:
3987:
3796:
3721:
3637:
3622:
3387:
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3131:
3126:
3023:
3013:
2985:
2723:
2665:
2012:
1768:
1617:
1605:
1581:
1306:
1167:
176:
2073:
1890:
706:{\displaystyle x\cdot a^{x-1}\cdot 0+a^{x}\cdot \left(1\cdot \log a+x\cdot {\frac {0}{a}}\right).}
4421:
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219:(usually different from the language used for the implementation), a dedicated memory manager, a
88:
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3711:
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2457:
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2246:
2223:
2182:
2127:
2117:
2092:
1970:
1894:
1827:
1817:
1463:
1163:
355:. There are also several other journals that regularly publish articles in computer algebra.
284:
in French, which means "formal computation". This name reflects the ties this field has with
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4197:
4045:
3982:
3614:
3498:
3465:
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2413:
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1962:
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919:
554:
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148:
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578:
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3939:
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3579:
3551:
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3222:
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2980:
2925:
2743:
2703:
1713:
1709:
1706:: an algorithm for dividing a polynomial by another polynomial of the same or lower degree
1537:
1533:
1363:
is the equality of their representation in a computer. This is easy to test in a program.
421:
409:
320:
1883:
2510:
Buchberger, Bruno; Collins, George Edwin; Loos, Rüdiger; Albrecht, Rudolf, eds. (1983).
2484:
1576:
at the
Massachusetts Institute of Technology and the organization that later became the
4368:
4295:
4002:
3791:
3695:
3594:
3440:
3412:
2512:
2473:
2344:
2079:. In Buchberger, Bruno; Collins, George Edwin; Loos, Rüdiger; Albrecht, Rudolf (eds.).
1621:
1600:
A large part of the work of researchers in the field consisted of revisiting classical
1541:
1529:
1495:
1471:
931:
398:
285:
220:
2417:
2282:
2202:"Recursive functions of symbolic expressions and their computation by machine, Part I"
534:; if they are not given any values, the result of the evaluation is simply its input.
4473:
4156:
4088:
4040:
3680:
2975:
1758:
1753:
923:
907:
4098:
4093:
3997:
3776:
3435:
2738:
2541:
1612:
for use in computer algebra. An example of this type of work is the computation of
417:
2437:. Translated from the French by A. Davenport and J. H. Davenport. Academic Press.
1712:: an algorithm for the calculus operation of indefinite integration (i.e. finding
17:
2305:
2038:(Master's thesis). Naval Postgraduate School, Monterey/CA. p. 15. ADA165184.
4300:
3964:
3887:
3766:
3392:
3304:
2356:
2351:, Computing Supplementa, vol. 4, Vienna: Springer Vienna, pp. 95–113,
2088:
1860:
1525:
436:
144:
459:
345:
There are several journals specializing in computer algebra, the top one being
80:
4285:
4164:
3959:
3786:
3716:
3309:
3049:
2905:
2789:
2523:
1966:
1691:
1482:
In computer algebra, "canonical form" and "normal form" are not synonymous. A
1459:
911:
566:
550:
252:
232:
2329:
2290:
2227:
2186:
2131:
1974:
1831:
3291:
3252:
2635:
2321:
2016:
1681:
1674:: find sums of hypergeometric terms that are themselves hypergeometric terms
720:
338:
There are several annual conferences on computer algebra, the premier being
164:
2548:
2178:
1155:
Some fundamental mathematical questions arise when one wants to manipulate
3855:
2218:
2201:
323:
that is specific to computer algebra, but this function is assumed by the
4189:
4108:
4035:
3352:
2779:
2660:
2645:
488:
402:
202:
168:
27:
Scientific area at the interface between computer science and mathematics
910:
of addition and multiplication. The problem is to quickly recognize the
3974:
2784:
2718:
2650:
2058:
1913:
1601:
915:
546:
272:
1367:
is when two expressions represent the same mathematical object, as in
2799:
2794:
2774:
2732:
2615:
2162:
1589:
927:
avoiding repetition of the same operations on identical expressions.
476:
332:
2492:
2842:
1951:"Élie Cartan's geometrical vision or how to avoid expression swell"
163:, is a scientific area that refers to the study and development of
2728:
2680:
2670:
2630:
2605:
2435:
Computer
Algebra: Systems and Algorithms for Algebraic Computation
2114:
Computer
Algebra: Systems and Algorithms for Algebraic Computation
1512:
413:
339:
48:
contains information that is not included elsewhere in the article
2001:"The Mathematica Kernel: Issues in the Design and Implementation"
1991:, from "Peculiarities of programming in computer algebra systems"
1632:) to a variant efficiently computable via a Euclidean algorithm.
335:(Special Interest Group on Symbolic and Algebraic Manipulation).
227:
to perform usual operations, like simplification of expressions,
223:
for the input/output of mathematical expressions, a large set of
2675:
1135:{\displaystyle (x-1)(x^{4}+x^{3}+x^{2}+x+1)\rightarrow x^{5}-1.}
291:
Symbolic computation has also been referred to, in the past, as
3859:
2846:
2552:
1885:
Computer
Algebra and Symbolic Computation: Mathematical Methods
738:. They are systematically applied in computer algebra systems.
1793:
179:. Although computer algebra could be considered a subfield of
29:
1030:{\displaystyle (x+1)^{4}\rightarrow x^{4}+4x^{3}+6x^{2}+4x+1}
1949:
Neut, Sylvain; Petitot, Michel; Dridi, Raouf (2009-03-01).
1620:, proved inefficient over infinite fields; algorithms from
205:
applications that perform symbolic calculations are called
1166:. This is not a real restriction, because, as soon as the
2151:. University of Pennsylvania. Retrieved December 3, 2023.
487:
may be viewed as the symbol of an operator followed by a
199:
that have no given value and are manipulated as symbols.
1844:
1668:: finds a Gröbner basis (also mentions the F5 algorithm)
1957:. Polynomial System Solving in honor of Daniel Lazard.
938:. For example, the distributivity law allows rewriting
2083:. Computing Supplementa. Vol. 4. pp. 11–43.
1376:
1309:
1271:
1179:
1043:
944:
611:
581:
2514:
Computer
Algebra: Symbolic and Algebraic Computation
2081:
Computer
Algebra: Symbolic and Algebraic Computation
1479:, and to test the syntactic equality of the result.
496:
be viewed as a program. For example, the expression
4359:
4316:
4226:
4188:
4155:
4107:
4079:
4026:
3973:
3930:
3704:
3651:
3613:
3560:
3522:
3484:
3426:
3343:
3289:
3251:
3203:
3140:
3073:
3037:
2994:
2958:
2891:
2757:
2689:
2586:
2511:
2472:
2112:Davenport, J. H.; Siret, Y.; Tournier, É. (1988).
1442:
1340:
1295:
1254:
1159:in a computer. We consider mainly the case of the
1134:
1029:
705:
594:
518:as parameters. Executing this program consists in
506:may be viewed as a program for the addition, with
376:, it is common, in computer algebra, to emphasize
2471:Geddes, K. O.; Czapor, S. R.; Labahn, G. (1992).
2452:von zur Gathen, Joachim; Gerhard, Jürgen (2003).
1255:{\displaystyle (\sin(x+y)^{2}+\log(z^{2}-5))^{3}}
2145:"ENIAC in Action: What it Was and How it Worked"
849:, the simplest way is to systematically rewrite
2051:Macsyma Mathematics and System Reference Manual
1810:Making Computer Algebra More Symbolic (Invited)
1564:for computing symbolic expressions through the
3871:
2882:Note: This template roughly follows the 2012
2858:
2564:
719:This simplification is normally done through
8:
2456:(2nd ed.). Cambridge University Press.
1511:Early computer algebra systems, such as the
2387:For a detailed definition of the subject:
2149:ENIAC: Celebrating Penn Engineering History
3878:
3864:
3856:
2865:
2851:
2843:
2825:
2571:
2557:
2549:
2013:"The GNU Multiple Precision (GMP) Library"
1443:{\displaystyle (x+y)^{2}=x^{2}+2xy+y^{2}.}
565:The raw application of the basic rules of
416:, usually the largest base allowed by the
2433:; Siret, Yvon; Tournier, Èvelyne (1988).
2217:
2072:Buchberger, Bruno; Loos, Rüdiger (1983).
1431:
1406:
1393:
1375:
1323:
1308:
1270:
1246:
1227:
1205:
1178:
1120:
1092:
1079:
1066:
1042:
1006:
990:
974:
961:
943:
685:
647:
622:
610:
586:
580:
68:Learn how and when to remove this message
458:
195:computation with expressions containing
191:, while symbolic computation emphasizes
79:
1785:
1662:: factor polynomials over finite fields
3575:Knowledge representation and reasoning
2518:. Computing Supplementa. Vol. 4.
2425:For textbooks devoted to the subject:
1355:There are two notions of equality for
3600:Philosophy of artificial intelligence
2397:"Symbolic Computation (An Editorial)"
2267:"Symbolic Computation (An Editorial)"
2003:. October 2006. Retrieved 2023-11-29.
1794:"ACM Association in computer algebra"
1643:List of algorithms § Computer algebra
1570:Massachusetts Institute of Technology
835:. In the case of expressions such as
420:. These integers allow to define the
7:
4440:
2926:Energy consumption (Green computing)
372:is highly efficient for approximate
4452:
3605:Distributed artificial intelligence
2884:ACM Computing Classification System
1636:Algorithms used in computer algebra
1614:polynomial greatest common divisors
906:Another difficulty occurs with the
522:the expression for given values of
329:Association for Computing Machinery
3117:Integrated development environment
1568:programming language while at the
1552:Foundations and early applications
393:The usual numbers systems used in
134:using the computer algebra system
25:
3585:Automated planning and scheduling
3122:Software configuration management
2304:Feldman, Stuart I. (1975-11-01).
1678:Knuth–Bendix completion algorithm
471:tree, from a 1985 Master's Thesis
463:Representation of the expression
435:and some commercial ones such as
4451:
4439:
4428:
4427:
4415:
3839:
3829:
3820:
3819:
2824:
2813:
2812:
2243:History of programming languages
1774:Symbolic artificial intelligence
1739:Computational algebraic geometry
1641:This section is an excerpt from
1475:or to put their difference in a
34:
4336:Computational complexity theory
3830:
3233:Computational complexity theory
2475:Algorithms for Computer Algebra
2405:Journal of Symbolic Computation
2306:"A brief description of Altran"
2271:Journal of Symbolic Computation
1955:Journal of Symbolic Computation
1688:Multivariate division algorithm
348:Journal of Symbolic Computation
3024:Network performance evaluation
2345:"Factorization of Polynomials"
2241:Wexelblat, Richard L. (1981).
2029:Cassidy, Kevin G. (Dec 1985).
1390:
1377:
1335:
1316:
1290:
1278:
1243:
1239:
1220:
1202:
1189:
1180:
1113:
1110:
1059:
1056:
1044:
967:
958:
945:
1:
3388:Multimedia information system
3373:Geographic information system
3363:Enterprise information system
2959:Computer systems organization
2418:10.1016/S0747-7171(85)80025-0
2283:10.1016/S0747-7171(85)80025-0
2200:McCarthy, John (1960-04-01).
1562:primitive recursive functions
1507:Human-driven computer algebra
1341:{\displaystyle \log(z^{2}-5)}
1265:is viewed as a polynomial in
3747:Computational social science
3335:Theoretical computer science
3155:Software development process
2931:Electronic design automation
2916:Very Large Scale Integration
1857:"SIGSAM list of conferences"
1764:Symbolic-numeric computation
1698:Pollard's kangaroo algorithm
3570:Natural language processing
3358:Information storage systems
2749:Engineering Equation Solver
2357:10.1007/978-3-7091-7551-4_8
2163:"When Computers Were Women"
2161:Light, Jennifer S. (1999).
2089:10.1007/978-3-7091-7551-4_2
1660:Cantor–Zassenhaus algorithm
1630:unique factorization domain
4496:
4386:Films about mathematicians
3486:Human–computer interaction
3456:Intrusion detection system
3368:Social information systems
3353:Database management system
2074:"Algebraic simplification"
1640:
1517:University of Pennsylvania
4409:
3955:Philosophy of mathematics
3895:
3815:
3752:Computational engineering
3727:Computational mathematics
2880:
2808:
2524:10.1007/978-3-7091-7551-4
2206:Communications of the ACM
1967:10.1016/j.jsc.2007.04.006
1807:Watt, Stephen M. (2006).
1694:in several indeterminates
1560:explored an extension of
1296:{\displaystyle \sin(x+y)}
741:A difficulty occurs with
263:Some authors distinguish
4391:Recreational mathematics
3762:Computational healthcare
3757:Differentiable computing
3676:Graphics processing unit
3102:Domain-specific language
2971:Computational complexity
2580:Computer algebra systems
1729:Automated theorem prover
1704:Polynomial long division
1357:mathematical expressions
1157:mathematical expressions
1145:Polynomial factorization
936:trigonometric identities
433:computer algebra systems
359:Computer science aspects
237:polynomial factorization
208:computer algebra systems
173:mathematical expressions
4276:Mathematical statistics
4266:Mathematical psychology
4236:Engineering mathematics
4170:Algebraic number theory
3737:Computational chemistry
3671:Photograph manipulation
3562:Artificial intelligence
3378:Decision support system
2454:Modern computer algebra
2322:10.1145/1088322.1088325
2019:. Retrieved 2023-11-29.
1914:SIGSAM list of journals
1881:Cohen, Joel S. (2003).
1744:Computer algebra system
1734:Computer-assisted proof
805:are both simplified to
485:mathematical expression
249:public key cryptography
4422:Mathematics portal
4271:Mathematical sociology
4251:Mathematical economics
4246:Mathematical chemistry
4175:Analytic number theory
4056:Differential equations
3802:Educational technology
3633:Reinforcement learning
3383:Process control system
3281:Computational geometry
3271:Algorithmic efficiency
3266:Analysis of algorithms
2921:Systems on Chip (SoCs)
2735:symbolic math toolbox)
2277:(1): 1–6. 1985-03-01.
2179:10.1353/tech.1999.0128
2167:Technology and Culture
1930:homepages.math.uic.edu
1651:Buchberger's algorithm
1578:Stanford AI Laboratory
1444:
1342:
1297:
1256:
1136:
1031:
743:associative operations
707:
596:
472:
325:special interest group
297:algebraic manipulation
241:indefinite integration
189:floating point numbers
140:
4401:Mathematics education
4331:Theory of computation
4051:Hypercomplex analysis
3772:Electronic publishing
3742:Computational biology
3732:Computational physics
3628:Unsupervised learning
3542:Distributed computing
3418:Information retrieval
3325:Mathematical analysis
3315:Mathematical software
3205:Theory of computation
3170:Software construction
3160:Requirements analysis
3038:Software organization
2966:Computer architecture
2936:Hardware acceleration
2901:Printed circuit board
2219:10.1145/367177.367199
1889:. AK Peters. p.
1749:Differential analyser
1445:
1343:
1298:
1257:
1137:
1032:
821:, which is displayed
708:
597:
595:{\displaystyle a^{x}}
462:
426:irreducible fractions
395:numerical computation
374:numerical computation
293:symbolic manipulation
185:numerical computation
161:algebraic computation
83:
4381:Informal mathematics
4261:Mathematical physics
4256:Mathematical finance
4241:Mathematical biology
4180:Diophantine geometry
3532:Concurrent computing
3504:Ubiquitous computing
3476:Application security
3471:Information security
3300:Discrete mathematics
3276:Randomized algorithm
3228:Computability theory
3213:Model of computation
3185:Software maintenance
3180:Software engineering
3142:Software development
3092:Programming language
3087:Programming paradigm
3004:Network architecture
2061:. 1996. p. 419.
1845:SIGSAM official site
1666:Faugère F4 algorithm
1610:efficient algorithms
1455:Richardson's theorem
1374:
1307:
1269:
1177:
1168:irrational functions
1151:Mathematical aspects
1041:
942:
609:
579:
315:Scientific community
305:symbolic mathematics
277:symbolic computation
269:symbolic computation
217:programming language
181:scientific computing
177:mathematical objects
157:symbolic computation
85:Symbolic integration
4396:Mathematics and art
4306:Operations research
4061:Functional analysis
3807:Document management
3797:Operations research
3722:Enterprise software
3638:Multi-task learning
3623:Supervised learning
3345:Information systems
3175:Software deployment
3132:Software repository
2986:Real-time computing
2485:1992afca.book.....G
2431:Davenport, James H.
2310:ACM SIGSAM Bulletin
1769:Symbolic simulation
1582:Stanford University
364:Data representation
351:founded in 1985 by
301:symbolic processing
275:. Some authors use
4341:Numerical analysis
3950:Mathematical logic
3945:Information theory
3590:Search methodology
3537:Parallel computing
3494:Interaction design
3403:Computing platform
3330:Numerical analysis
3320:Information theory
3112:Software framework
3075:Software notations
3014:Network components
2911:Integrated circuit
1672:Gosper's algorithm
1618:Euclid's algorithm
1548:led said efforts.
1464:rational fractions
1440:
1361:Syntactic equality
1338:
1293:
1252:
1164:rational fractions
1132:
1027:
876:as, respectively,
759:is represented as
703:
602:gives the result
592:
575:on the expression
549:) or entries in a
473:
447:, which is thus a
414:base of numeration
370:numerical software
141:
89:algebraic function
18:Syntactic equality
4467:
4466:
4066:Harmonic analysis
3853:
3852:
3782:Electronic voting
3712:Quantum Computing
3705:Applied computing
3691:Image compression
3461:Hardware security
3451:Security services
3408:Digital marketing
3195:Open-source model
3107:Modeling language
3019:Network scheduler
2840:
2839:
2533:978-3-211-81776-6
2502:978-0-7923-9259-0
2444:978-0-12-204230-0
2393:Buchberger, Bruno
2366:978-3-211-81776-6
2252:978-0-12-745040-7
2098:978-3-211-81776-6
1900:978-1-56881-159-8
1608:while developing
1596:Historic problems
1453:It is known from
1365:Semantic equality
693:
465:(8 − 6) × (3 + 1)
428:of two integers.
187:with approximate
171:for manipulating
78:
77:
70:
16:(Redirected from
4487:
4480:Computer algebra
4455:
4454:
4443:
4442:
4431:
4430:
4420:
4419:
4351:Computer algebra
4326:Computer science
4046:Complex analysis
3880:
3873:
3866:
3857:
3843:
3842:
3833:
3832:
3823:
3822:
3643:Cross-validation
3615:Machine learning
3499:Social computing
3466:Network security
3261:Algorithm design
3190:Programming team
3150:Control variable
3127:Software library
3065:Software quality
3060:Operating system
3009:Network protocol
2874:Computer science
2867:
2860:
2853:
2844:
2828:
2827:
2816:
2815:
2699:ClassPad Manager
2573:
2566:
2559:
2550:
2545:
2517:
2506:
2478:
2467:
2448:
2421:
2401:
2376:
2375:
2374:
2373:
2349:Computer Algebra
2340:
2334:
2333:
2301:
2295:
2294:
2263:
2257:
2256:
2238:
2232:
2231:
2221:
2197:
2191:
2190:
2158:
2152:
2142:
2136:
2135:
2109:
2103:
2102:
2078:
2069:
2063:
2062:
2056:
2046:
2040:
2039:
2037:
2026:
2020:
2010:
2004:
1998:
1992:
1989:Expression swell
1985:
1979:
1978:
1946:
1940:
1939:
1937:
1936:
1922:
1916:
1911:
1905:
1904:
1888:
1878:
1872:
1871:
1869:
1868:
1859:. Archived from
1853:
1847:
1842:
1836:
1835:
1815:
1804:
1798:
1797:
1790:
1626:ring of integers
1604:to increase its
1449:
1447:
1446:
1441:
1436:
1435:
1411:
1410:
1398:
1397:
1347:
1345:
1344:
1339:
1328:
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1210:
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1141:
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1125:
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1097:
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1084:
1083:
1071:
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1036:
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1033:
1028:
1011:
1010:
995:
994:
979:
978:
966:
965:
902:
892:
882:
875:
865:
855:
848:
834:
820:
804:
789:
774:
758:
737:
733:
712:
710:
709:
704:
699:
695:
694:
686:
652:
651:
633:
632:
601:
599:
598:
593:
591:
590:
574:
569:with respect to
533:
527:
517:
511:
505:
466:
422:rational numbers
383:expression swell
353:Bruno Buchberger
309:symbolic algebra
265:computer algebra
211:, with the term
153:computer algebra
149:computer science
133:
132:
130:
129:
128:
127:
110:
107:
73:
66:
62:
59:
53:
38:
37:
30:
21:
4495:
4494:
4490:
4489:
4488:
4486:
4485:
4484:
4470:
4469:
4468:
4463:
4414:
4405:
4355:
4312:
4291:Systems science
4222:
4218:Homotopy theory
4184:
4151:
4103:
4075:
4022:
3969:
3940:Category theory
3926:
3891:
3884:
3854:
3849:
3840:
3811:
3792:Word processing
3700:
3686:Virtual reality
3647:
3609:
3580:Computer vision
3556:
3552:Multiprocessing
3518:
3480:
3446:Security hacker
3422:
3398:Digital library
3339:
3290:Mathematics of
3285:
3247:
3223:Automata theory
3218:Formal language
3199:
3165:Software design
3136:
3069:
3055:Virtual machine
3033:
3029:Network service
2990:
2981:Embedded system
2954:
2887:
2876:
2871:
2841:
2836:
2804:
2753:
2744:TI InterActive!
2685:
2582:
2577:
2534:
2509:
2503:
2493:10.1007/b102438
2470:
2464:
2451:
2445:
2429:
2399:
2391:
2385:
2383:Further reading
2380:
2379:
2371:
2369:
2367:
2342:
2341:
2337:
2303:
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2298:
2265:
2264:
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2253:
2240:
2239:
2235:
2199:
2198:
2194:
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2066:
2054:
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2028:
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2011:
2007:
1999:
1995:
1986:
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1839:
1824:
1813:
1806:
1805:
1801:
1792:
1791:
1787:
1782:
1725:
1720:
1719:
1714:antiderivatives
1710:Risch algorithm
1646:
1638:
1598:
1554:
1534:Ruth Lichterman
1521:human computers
1509:
1504:
1427:
1402:
1389:
1372:
1371:
1353:
1319:
1305:
1304:
1267:
1266:
1242:
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1175:
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1038:
1002:
986:
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957:
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939:
894:
884:
877:
867:
857:
850:
836:
822:
806:
791:
776:
760:
746:
735:
724:
721:rewriting rules
660:
656:
643:
618:
607:
606:
582:
577:
576:
570:
567:differentiation
563:
529:
523:
513:
507:
497:
464:
457:
391:
366:
361:
321:learned society
317:
261:
229:differentiation
114:
112:
111:
108:
103:
102:
100:
91:
74:
63:
57:
54:
51:
43:This article's
39:
35:
28:
23:
22:
15:
12:
11:
5:
4493:
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4425:
4410:
4407:
4406:
4404:
4403:
4398:
4393:
4388:
4383:
4378:
4377:
4376:
4369:Mathematicians
4365:
4363:
4361:Related topics
4357:
4356:
4354:
4353:
4348:
4343:
4338:
4333:
4328:
4322:
4320:
4314:
4313:
4311:
4310:
4309:
4308:
4303:
4298:
4296:Control theory
4288:
4283:
4278:
4273:
4268:
4263:
4258:
4253:
4248:
4243:
4238:
4232:
4230:
4224:
4223:
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4205:
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4167:
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4150:
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4144:
4139:
4134:
4129:
4124:
4119:
4113:
4111:
4105:
4104:
4102:
4101:
4096:
4091:
4085:
4083:
4077:
4076:
4074:
4073:
4071:Measure theory
4068:
4063:
4058:
4053:
4048:
4043:
4038:
4032:
4030:
4024:
4023:
4021:
4020:
4015:
4010:
4005:
4000:
3995:
3990:
3985:
3979:
3977:
3971:
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3962:
3957:
3952:
3947:
3942:
3936:
3934:
3928:
3927:
3925:
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3907:
3896:
3893:
3892:
3885:
3883:
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3868:
3860:
3851:
3850:
3848:
3847:
3837:
3827:
3816:
3813:
3812:
3810:
3809:
3804:
3799:
3794:
3789:
3784:
3779:
3774:
3769:
3764:
3759:
3754:
3749:
3744:
3739:
3734:
3729:
3724:
3719:
3714:
3708:
3706:
3702:
3701:
3699:
3698:
3696:Solid modeling
3693:
3688:
3683:
3678:
3673:
3668:
3663:
3657:
3655:
3649:
3648:
3646:
3645:
3640:
3635:
3630:
3625:
3619:
3617:
3611:
3610:
3608:
3607:
3602:
3597:
3595:Control method
3592:
3587:
3582:
3577:
3572:
3566:
3564:
3558:
3557:
3555:
3554:
3549:
3547:Multithreading
3544:
3539:
3534:
3528:
3526:
3520:
3519:
3517:
3516:
3511:
3506:
3501:
3496:
3490:
3488:
3482:
3481:
3479:
3478:
3473:
3468:
3463:
3458:
3453:
3448:
3443:
3441:Formal methods
3438:
3432:
3430:
3424:
3423:
3421:
3420:
3415:
3413:World Wide Web
3410:
3405:
3400:
3395:
3390:
3385:
3380:
3375:
3370:
3365:
3360:
3355:
3349:
3347:
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3338:
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3327:
3322:
3317:
3312:
3307:
3302:
3296:
3294:
3287:
3286:
3284:
3283:
3278:
3273:
3268:
3263:
3257:
3255:
3249:
3248:
3246:
3245:
3240:
3235:
3230:
3225:
3220:
3215:
3209:
3207:
3201:
3200:
3198:
3197:
3192:
3187:
3182:
3177:
3172:
3167:
3162:
3157:
3152:
3146:
3144:
3138:
3137:
3135:
3134:
3129:
3124:
3119:
3114:
3109:
3104:
3099:
3094:
3089:
3083:
3081:
3071:
3070:
3068:
3067:
3062:
3057:
3052:
3047:
3041:
3039:
3035:
3034:
3032:
3031:
3026:
3021:
3016:
3011:
3006:
3000:
2998:
2992:
2991:
2989:
2988:
2983:
2978:
2973:
2968:
2962:
2960:
2956:
2955:
2953:
2952:
2943:
2938:
2933:
2928:
2923:
2918:
2913:
2908:
2903:
2897:
2895:
2889:
2888:
2881:
2878:
2877:
2872:
2870:
2869:
2862:
2855:
2847:
2838:
2837:
2835:
2834:
2822:
2809:
2806:
2805:
2803:
2802:
2797:
2792:
2787:
2782:
2777:
2772:
2767:
2761:
2759:
2755:
2754:
2752:
2751:
2746:
2741:
2736:
2726:
2721:
2716:
2711:
2706:
2701:
2695:
2693:
2687:
2686:
2684:
2683:
2678:
2673:
2668:
2663:
2658:
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2648:
2643:
2638:
2633:
2628:
2623:
2618:
2613:
2608:
2603:
2598:
2592:
2590:
2584:
2583:
2578:
2576:
2575:
2568:
2561:
2553:
2547:
2546:
2532:
2507:
2501:
2468:
2462:
2449:
2443:
2423:
2422:
2384:
2381:
2378:
2377:
2365:
2335:
2296:
2258:
2251:
2233:
2212:(4): 184–195.
2192:
2173:(3): 455–483.
2153:
2137:
2122:
2104:
2097:
2064:
2041:
2021:
2005:
1993:
1987:Richard Liska
1980:
1961:(3): 261–270.
1941:
1917:
1906:
1899:
1873:
1848:
1837:
1822:
1799:
1784:
1783:
1781:
1778:
1777:
1776:
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1766:
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1756:
1751:
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1741:
1736:
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1724:
1721:
1718:
1717:
1707:
1701:
1695:
1685:
1675:
1669:
1663:
1657:
1647:
1639:
1637:
1634:
1622:linear algebra
1597:
1594:
1553:
1550:
1530:Marlyn Wescoff
1508:
1505:
1503:
1500:
1496:distributivity
1484:canonical form
1472:canonical form
1451:
1450:
1439:
1434:
1430:
1426:
1423:
1420:
1417:
1414:
1409:
1405:
1401:
1396:
1392:
1388:
1385:
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1379:
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1322:
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1082:
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1005:
1001:
998:
993:
989:
985:
982:
977:
973:
969:
964:
960:
956:
953:
950:
947:
932:distributivity
714:
713:
702:
698:
692:
689:
684:
681:
678:
675:
672:
669:
666:
663:
659:
655:
650:
646:
642:
639:
636:
631:
628:
625:
621:
617:
614:
589:
585:
562:
561:Simplification
559:
456:
453:
399:floating point
390:
387:
365:
362:
360:
357:
316:
313:
286:formal methods
260:
257:
251:, or for some
221:user interface
155:, also called
76:
75:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4492:
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4477:
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2976:Dependability
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2463:0-521-82646-2
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2123:0-12-204230-1
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1863:on 2013-08-08
1862:
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1823:9788468983813
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1759:Model checker
1757:
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1754:Proof checker
1752:
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1658:
1656:
1655:Gröbner basis
1652:
1649:
1648:
1644:
1635:
1633:
1631:
1627:
1623:
1619:
1615:
1611:
1607:
1606:effectiveness
1603:
1595:
1593:
1591:
1585:
1583:
1579:
1575:
1571:
1567:
1563:
1559:
1558:John McCarthy
1551:
1549:
1547:
1543:
1542:Frances Bilas
1539:
1535:
1531:
1527:
1526:Jean Jennings
1522:
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948:
937:
933:
928:
925:
924:hash function
921:
917:
913:
909:
908:commutativity
904:
901:
897:
891:
888:+ (−1)⋅
887:
881:
874:
870:
864:
860:
854:
847:
843:
839:
833:
829:
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336:
334:
330:
326:
322:
314:
312:
310:
306:
302:
298:
294:
289:
287:
283:
282:calcul formel
278:
274:
270:
266:
258:
256:
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250:
244:
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238:
234:
230:
226:
222:
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139:
138:
125:
121:
117:
106:
98:
94:
90:
86:
82:
72:
69:
61:
49:
47:
41:
32:
31:
19:
4456:
4444:
4432:
4413:
4350:
4346:Optimization
4208:Differential
4132:Differential
4099:Order theory
4094:Graph theory
3998:Group theory
3777:Cyberwarfare
3436:Cryptography
2829:
2817:
2758:Discontinued
2739:SMath Studio
2513:
2474:
2453:
2434:
2424:
2409:
2403:
2386:
2370:, retrieved
2348:
2338:
2316:(4): 12–20.
2313:
2309:
2299:
2274:
2270:
2261:
2242:
2236:
2209:
2205:
2195:
2170:
2166:
2156:
2148:
2140:
2116:. Academic.
2113:
2107:
2080:
2067:
2050:
2044:
2031:
2024:
2008:
1996:
1983:
1958:
1954:
1944:
1933:. Retrieved
1929:
1920:
1909:
1884:
1876:
1865:. Retrieved
1861:the original
1851:
1840:
1809:
1802:
1788:
1684:rule systems
1599:
1586:
1555:
1538:Betty Snyder
1519:, relied on
1510:
1492:
1487:
1483:
1481:
1476:
1470:
1468:
1452:
1364:
1360:
1354:
1264:
1161:multivariate
1154:
929:
905:
899:
895:
889:
885:
879:
872:
868:
862:
858:
852:
845:
841:
837:
831:
827:
823:
816:
812:
808:
801:
797:
793:
785:
781:
777:
770:
766:
762:
755:
751:
747:
740:
729:
725:
718:
715:
571:
564:
540:
536:
530:
524:
519:
514:
508:
502:
498:
494:
474:
448:
430:
424:, which are
418:machine word
407:
401:numbers and
392:
382:
377:
367:
346:
344:
337:
319:There is no
318:
308:
304:
300:
296:
292:
290:
281:
276:
268:
264:
262:
245:
212:
206:
201:
192:
160:
156:
152:
142:
135:
123:
119:
115:
104:
96:
92:
64:
55:
46:lead section
44:
4458:WikiProject
4301:Game theory
4281:Probability
4018:Homological
4008:Multilinear
3988:Commutative
3965:Type theory
3932:Foundations
3888:mathematics
3787:Video games
3767:Digital art
3524:Concurrency
3393:Data mining
3305:Probability
3045:Interpreter
2724:Mathematica
2691:Proprietary
2588:Open-source
1692:polynomials
1574:Project MAC
1546:Kay McNulty
1488:normal form
1477:normal form
1460:polynomials
878:(−1)⋅
475:Except for
455:Expressions
445:GMP library
437:Mathematica
259:Terminology
145:mathematics
4286:Statistics
4165:Arithmetic
4127:Arithmetic
3993:Elementary
3960:Set theory
3845:Glossaries
3717:E-commerce
3310:Statistics
3253:Algorithms
3050:Middleware
2906:Peripheral
2790:Mathomatic
2412:(1): 1–6.
2372:2023-11-29
1935:2024-03-31
1867:2012-11-15
1780:References
1653:: finds a
1580:(SAIL) at
912:like terms
736:sin(0) → 0
551:hash table
520:evaluating
451:standard.
443:, use the
255:problems.
253:non-linear
233:chain rule
231:using the
175:and other
165:algorithms
4213:Geometric
4203:Algebraic
4142:Euclidean
4117:Algebraic
4013:Universal
3666:Rendering
3661:Animation
3292:computing
3243:Semantics
2941:Processor
2676:Xcas/Giac
2636:Macaulay2
2330:0163-5824
2291:0747-7171
2228:0001-0782
2187:1097-3729
2132:802584470
2017:Maplesoft
1975:0747-7171
1832:496720771
1682:rewriting
1556:In 1960,
1330:−
1314:
1276:
1234:−
1218:
1187:
1127:−
1114:→
1051:−
968:→
683:⋅
671:
665:⋅
654:⋅
635:⋅
627:−
616:⋅
553:(like in
545:(like in
481:variables
197:variables
4474:Category
4434:Category
4190:Topology
4137:Discrete
4122:Analytic
4109:Geometry
4081:Discrete
4036:Calculus
4028:Analysis
3983:Abstract
3922:Glossary
3905:Timeline
3825:Category
3653:Graphics
3428:Security
3097:Compiler
2996:Networks
2893:Hardware
2819:Category
2780:LiveMath
2666:Singular
2661:SageMath
2646:Normaliz
2395:(1985).
1723:See also
1351:Equality
861:−
840:−
728:−
543:pointers
489:sequence
483:, every
449:de facto
412:in some
403:integers
273:formulas
225:routines
203:Software
169:software
58:May 2020
4446:Commons
4228:Applied
4198:General
3975:Algebra
3900:History
3835:Outline
2785:Macsyma
2719:Mathcad
2651:PARI/GP
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