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{
    "compare": {
        "fromid": 1,
        "fromrevid": 1,
        "fromns": 0,
        "fromtitle": "Alg\u00e8bre lin\u00e9aire",
        "toid": 2,
        "torevid": 2,
        "tons": 0,
        "totitle": "Algorithmique",
        "*": "<tr><td colspan=\"2\" class=\"diff-lineno\" id=\"mw-diff-left-l1\">Ligne\u00a01\u202f:</td>\n<td colspan=\"2\" class=\"diff-lineno\">Ligne\u00a01\u202f:</td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">{{Voir homonymes</del>|<del class=\"diffchange diffchange-inline\">Alg\u00e8bre (homonymie)}}</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Fichier:Euclid flowchart 1.png</ins>|<ins class=\"diffchange diffchange-inline\">vignette</ins>|<ins class=\"diffchange diffchange-inline\">[[Organigramme </ins>de <ins class=\"diffchange diffchange-inline\">programmation]] repr\u00e9sentant l'</ins>[[<ins class=\"diffchange diffchange-inline\">algorithme d'Euclide</ins>]].<ins class=\"diffchange diffchange-inline\">]]</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">{{Infobox Discipline</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>|<del class=\"diffchange diffchange-inline\">partie </del>de<del class=\"diffchange diffchange-inline\">=</del>[[<del class=\"diffchange diffchange-inline\">Alg\u00e8bre</del>]]<del class=\"diffchange diffchange-inline\">|image=Linear subspaces with shading</del>.<del class=\"diffchange diffchange-inline\">svg|l\u00e9gende=</del>[[<del class=\"diffchange diffchange-inline\">\u211d\u00b3</del>]] est un <del class=\"diffchange diffchange-inline\">espace vectoriel </del>de <del class=\"diffchange diffchange-inline\">dimension 3. Les droites et plans qui passent par l'origine sont </del>des <del class=\"diffchange diffchange-inline\">sous-espaces vectoriels</del>. \u00a0</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">L{{'}}'''algorithmique''' est l'\u00e9tude et la production de r\u00e8gles et techniques qui sont impliqu\u00e9es dans la d\u00e9finition et la conception d'</ins>[[<ins class=\"diffchange diffchange-inline\">algorithme</ins>]]<ins class=\"diffchange diffchange-inline\">s, c'</ins>est<ins class=\"diffchange diffchange-inline\">-\u00e0-dire de processus syst\u00e9matiques de r\u00e9solution d'</ins>un <ins class=\"diffchange diffchange-inline\">probl\u00e8me permettant </ins>de <ins class=\"diffchange diffchange-inline\">d\u00e9crire pr\u00e9cis\u00e9ment </ins>des <ins class=\"diffchange diffchange-inline\">\u00e9tapes pour r\u00e9soudre un [[probl\u00e8me algorithmique]]</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">}}</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">L\u2019</del>'''<del class=\"diffchange diffchange-inline\">alg\u00e8bre lin\u00e9aire</del>'<del class=\"diffchange diffchange-inline\">'' est </del>la <del class=\"diffchange diffchange-inline\">branche </del>des [[<del class=\"diffchange diffchange-inline\">math\u00e9matiques</del>]] <del class=\"diffchange diffchange-inline\">qui s'int\u00e9resse aux </del>[[<del class=\"diffchange diffchange-inline\">Espace vectoriel</del>|<del class=\"diffchange diffchange-inline\">espaces vectoriels</del>]] et <del class=\"diffchange diffchange-inline\">aux </del>[[<del class=\"diffchange diffchange-inline\">Application lin\u00e9aire</del>|<del class=\"diffchange diffchange-inline\">transformations lin\u00e9aires</del>]]<del class=\"diffchange diffchange-inline\">, formalisation g\u00e9n\u00e9rale des th\u00e9ories des </del>[[<del class=\"diffchange diffchange-inline\">Syst\u00e8me d</del>'<del class=\"diffchange diffchange-inline\">\u00e9quations lin\u00e9aires</del>|<del class=\"diffchange diffchange-inline\">syst\u00e8mes d</del>'<del class=\"diffchange diffchange-inline\">\u00e9quations lin\u00e9aires</del>]].</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== \u00c9tymologie ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Le mot \u00ab algorithme \u00bb vient du nom du [[math\u00e9maticien]] [[Al-Khw\u00e2rizm\u00ee]]&lt;ref&gt;{{Lien web|auteur1=Phillipe Collard|auteur2=[[Philippe Flajolet]]|titre=Algorithmique|url=http://www.universalis.fr/encyclopedie/algorithmique/|site=Encyclop\u00e6dia universalis|consult\u00e9 le=8 mars 2015}}.&lt;/ref&gt; (latinis\u00e9 au [[Moyen \u00c2ge]] en {{lang|la|</ins>''<ins class=\"diffchange diffchange-inline\">Algoritmi</ins>''<ins class=\"diffchange diffchange-inline\">}}), qui, au {{IXe si\u00e8cle}} \u00e9crivit [[Abr\u00e9g\u00e9 du calcul par la restauration et </ins>la <ins class=\"diffchange diffchange-inline\">comparaison|le premier ouvrage syst\u00e9matique]] donnant </ins>des <ins class=\"diffchange diffchange-inline\">solutions aux </ins>[[<ins class=\"diffchange diffchange-inline\">\u00e9quation lin\u00e9aire|\u00e9quations lin\u00e9aires</ins>]] <ins class=\"diffchange diffchange-inline\">et </ins>[[<ins class=\"diffchange diffchange-inline\">\u00e9quation du second degr\u00e9</ins>|<ins class=\"diffchange diffchange-inline\">quadratiques</ins>]]<ins class=\"diffchange diffchange-inline\">. Le h muet, non justifi\u00e9 par l'\u00e9tymologie, vient d\u2019une d\u00e9formation par rapprochement avec le grec {{lang|el|\u1f00\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2}} (arithm\u00f3s)&lt;ref&gt;Albert Dauzat, Jean Dubois, Henri Mitterand, ''Nouveau dictionnaire \u00e9tymologique </ins>et <ins class=\"diffchange diffchange-inline\">historique'', 1971&lt;/ref&gt;. \u00ab Algorithme \u00bb a donn\u00e9 \u00ab algorithmique \u00bb. Le synonyme \u00ab algorithmie \u00bb, vieux mot utilis\u00e9 par exemple par </ins>[[<ins class=\"diffchange diffchange-inline\">Josef Ho\u00ebn\u00e9-Wronski</ins>|<ins class=\"diffchange diffchange-inline\">Wronski</ins>]] <ins class=\"diffchange diffchange-inline\">en 1811&lt;ref&gt;{{Ouvrage|auteur1=</ins>[[<ins class=\"diffchange diffchange-inline\">Josef Ho\u00ebn\u00e9-Wronski|Ho\u00e9n\u00e9 de Wronski]]|titre=Introduction \u00e0 la philosophie des math\u00e9matiques et technie de l</ins>'<ins class=\"diffchange diffchange-inline\">algorithmie|\u00e9diteur=Chez Courcier, imprimeur-libraire pour les math\u00e9matiques|ann\u00e9e=1811</ins>|<ins class=\"diffchange diffchange-inline\">lire en ligne=https://gallica.bnf.fr/ark:/12148/bpt6k6225961k.r=algorithmie}}&lt;/ref&gt;, est encore parfois utilis\u00e9&lt;ref&gt;Par exemple, l</ins>'<ins class=\"diffchange diffchange-inline\">[[Universit\u00e9 du Qu\u00e9bec \u00e0 Montr\u00e9al|UQAM]] propose un cours intitul\u00e9 \u00ab [https://etudier.uqam.ca/cours?sigle=EDM4600 Algorithmie de base et interactivit\u00e9</ins>] <ins class=\"diffchange diffchange-inline\">\u00bb, et l'universit\u00e9 de Montr\u00e9al, un cours intitul\u00e9 \u00ab [https://studium.umontreal.ca/course/info.php?id=48463 Algorithmie et effets audionum\u00e9riques</ins>] <ins class=\"diffchange diffchange-inline\">\u00bb.&lt;/ref&gt;</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><div>== Histoire ==</div></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><div>== Histoire ==</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">L</del>'<del class=\"diffchange diffchange-inline\">alg\u00e8bre lin\u00e9aire est initi\u00e9e dans son principe par le math\u00e9maticien perse </del>[[<del class=\"diffchange diffchange-inline\">Al-Khw\u00e2rizm\u00ee</del>]] <del class=\"diffchange diffchange-inline\">qui s'est inspir\u00e9 </del>des <del class=\"diffchange diffchange-inline\">textes </del>de math\u00e9matiques <del class=\"diffchange diffchange-inline\">indiens </del>et <del class=\"diffchange diffchange-inline\">qui a compl\u00e9t\u00e9 les travaux de </del>l'<del class=\"diffchange diffchange-inline\">\u00e9cole grecque, laquelle continuera de se d\u00e9velopper des si\u00e8cles durant</del>&lt;ref&gt;[[<del class=\"diffchange diffchange-inline\">Roshdi Rashed</del>]], ''<del class=\"diffchange diffchange-inline\">D</del>'<del class=\"diffchange diffchange-inline\">Al Khwarizmi \u00e0 Descartes, \u00c9tude sur l</del>'histoire <del class=\"diffchange diffchange-inline\">des math\u00e9matiques classiques'</del>', <del class=\"diffchange diffchange-inline\">Hermann, 2011</del>&lt;/ref&gt;. <del class=\"diffchange diffchange-inline\">Elle a \u00e9t\u00e9 reprise par </del>[[<del class=\"diffchange diffchange-inline\">Ren\u00e9 Descartes</del>]] <del class=\"diffchange diffchange-inline\">qui pose des probl\u00e8mes </del>de [[<del class=\"diffchange diffchange-inline\">g\u00e9om\u00e9trie</del>]], <del class=\"diffchange diffchange-inline\">comme la d\u00e9termination de l'intersection </del>de deux [[<del class=\"diffchange diffchange-inline\">Droite (math\u00e9matiques)</del>|<del class=\"diffchange diffchange-inline\">droites</del>]]<del class=\"diffchange diffchange-inline\">, en termes d</del>'[[<del class=\"diffchange diffchange-inline\">\u00e9quation lin\u00e9aire</del>]]<del class=\"diffchange diffchange-inline\">, \u00e9tablissant d\u00e8s lors </del>un <del class=\"diffchange diffchange-inline\">pont entre deux branches math\u00e9matiques jusqu</del>'<del class=\"diffchange diffchange-inline\">alors s\u00e9par\u00e9es </del>: l<del class=\"diffchange diffchange-inline\">'</del>[[<del class=\"diffchange diffchange-inline\">alg\u00e8bre</del>]] et <del class=\"diffchange diffchange-inline\">la g\u00e9om\u00e9trie</del>. <del class=\"diffchange diffchange-inline\">S</del>'<del class=\"diffchange diffchange-inline\">il ne d\u00e9finit pas </del>la <del class=\"diffchange diffchange-inline\">notion de base de l</del>'<del class=\"diffchange diffchange-inline\">alg\u00e8bre lin\u00e9aire qu</del>'<del class=\"diffchange diffchange-inline\">est celle d'espace vectoriel</del>, il <del class=\"diffchange diffchange-inline\">l'</del>utilise <del class=\"diffchange diffchange-inline\">d\u00e9j\u00e0 avec succ\u00e8s, et cette utilisation naturelle </del>des <del class=\"diffchange diffchange-inline\">aspects lin\u00e9aires </del>des <del class=\"diffchange diffchange-inline\">\u00e9quations manipul\u00e9es demeurera utilis\u00e9e de mani\u00e8re ''ad hoc''</del>, <del class=\"diffchange diffchange-inline\">fond\u00e9e essentiellement sur </del>les <del class=\"diffchange diffchange-inline\">id\u00e9es g\u00e9om\u00e9triques sous</del>-<del class=\"diffchange diffchange-inline\">jacentes. Apr\u00e8s cette d\u00e9couverte, les progr\u00e8s en alg\u00e8bre lin\u00e9aire vont se limiter \u00e0 des \u00e9tudes ponctuelles comme la d\u00e9finition et l'analyse des premi\u00e8res propri\u00e9t\u00e9s des </del>[[<del class=\"diffchange diffchange-inline\">d\u00e9terminant </del>(<del class=\"diffchange diffchange-inline\">math\u00e9matiques</del>)<del class=\"diffchange diffchange-inline\">|d\u00e9terminants</del>]] par [[<del class=\"diffchange diffchange-inline\">Jean Le Rond d</del>'<del class=\"diffchange diffchange-inline\">Alembert|Jean d</del>'<del class=\"diffchange diffchange-inline\">Alembert</del>]].</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[File:Cuneiform tablet- fragment of a mathematical problem text MET ME86 11 404.jpg|thumb|Fragment d</ins>'<ins class=\"diffchange diffchange-inline\">une tablette cun\u00e9iforme avec un probl\u00e8me algorithmique. MET ME86 11 404]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== Antiquit\u00e9 ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Les premiers algorithmes dont on a retrouv\u00e9 des descriptions datent des </ins>[[<ins class=\"diffchange diffchange-inline\">Babylone|Babyloniens</ins>]]<ins class=\"diffchange diffchange-inline\">, au {{-mill\u00e9naire|III|e}}. Ils d\u00e9crivent </ins>des <ins class=\"diffchange diffchange-inline\">m\u00e9thodes </ins>de <ins class=\"diffchange diffchange-inline\">[[Calcul (</ins>math\u00e9matiques<ins class=\"diffchange diffchange-inline\">)|calcul]] </ins>et <ins class=\"diffchange diffchange-inline\">des r\u00e9solutions d'[[\u00c9quation|\u00e9quations]] \u00e0 </ins>l'<ins class=\"diffchange diffchange-inline\">aide d'exemples</ins>&lt;ref&gt;<ins class=\"diffchange diffchange-inline\">{{article|auteur=[[Donald Knuth]]|p\u00e9riodique=[[Communications of the ACM]]|titre=Ancient Babylonian Algorithms|volume=15|num\u00e9ro=7|date=juillet 1972}}, repris dans {{Ouvrage|auteur1=</ins>[[<ins class=\"diffchange diffchange-inline\">Donald Knuth</ins>]]<ins class=\"diffchange diffchange-inline\">|titre=Selected Papers on Computer Science|\u00e9diteur=[[Addison-Wesley]]|ann\u00e9e=1996|passage=185}}</ins>, <ins class=\"diffchange diffchange-inline\">traduit en fran\u00e7ais sous le titre </ins>''<ins class=\"diffchange diffchange-inline\">Algoritmes babyloniens anciens</ins>'' <ins class=\"diffchange diffchange-inline\">dans {{Ouvrage|langue=fr|auteur1=[[Donald Knuth]]|traducteur=P. C\u00e9gielski|titre=\u00c9l\u00e9ments pour une </ins>histoire <ins class=\"diffchange diffchange-inline\">de l</ins>'<ins class=\"diffchange diffchange-inline\">informatique|\u00e9diteur=[[Librairie Eyrolles]]|ann\u00e9e=2011}}.&lt;/ref&gt;{{</ins>,<ins class=\"diffchange diffchange-inline\">}}&lt;ref&gt;{{article| auteur = Christine Proust | titre = Math\u00e9matiques en M\u00e9sopotamie | p\u00e9riodique = Images des Math\u00e9matiques| date = 14 avril 2014| lire en ligne = http://images.math.cnrs.fr/Mathematiques-en-Mesopotamie}}.</ins>&lt;/ref&gt;.</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Un algorithme c\u00e9l\u00e8bre est celui qui se trouve dans le {{nobr|livre 7}} des ''[[algorithme d'Euclide|\u00c9l\u00e9ments d'Euclide]]'', et appel\u00e9 </ins>[[<ins class=\"diffchange diffchange-inline\">algorithme d'Euclide</ins>]]<ins class=\"diffchange diffchange-inline\">. Il permet </ins>de <ins class=\"diffchange diffchange-inline\">trouver le plus grand diviseur commun, ou </ins>[[<ins class=\"diffchange diffchange-inline\">Plus grand commun diviseur|PGCD</ins>]], de deux <ins class=\"diffchange diffchange-inline\">nombres. Un point particuli\u00e8rement remarquable est qu\u2019il contient explicitement une </ins>[[<ins class=\"diffchange diffchange-inline\">it\u00e9ration]] et que les {{nobr|propositions 1}} et 2 d\u00e9montrent sa [[correction d'un algorithme</ins>|<ins class=\"diffchange diffchange-inline\">correction</ins>]]<ins class=\"diffchange diffchange-inline\">.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">C</ins>'<ins class=\"diffchange diffchange-inline\">est </ins>[[<ins class=\"diffchange diffchange-inline\">Archim\u00e8de</ins>]] <ins class=\"diffchange diffchange-inline\">qui proposa le premier </ins>un <ins class=\"diffchange diffchange-inline\">algorithme pour le calcul de {{math|[[Pi|\u03c0]]}}&lt;ref&gt;Le calcul de {{math|\u03c0}} {{citation|est caract\u00e9ristique des probl\u00e8mes g\u00e9n\u00e9raux rencontr\u00e9s en algorithmique.}} {{Lien web|auteur1=Phillipe Collard|auteur2=Phillipe Flajolet|titre=Algorithmique|sous-titre=1. L</ins>'<ins class=\"diffchange diffchange-inline\">exemple du calcul de {{math|\u03c0}}|url=http</ins>:<ins class=\"diffchange diffchange-inline\">//www.universalis.fr/encyclopedie/algorithmique/1-</ins>l<ins class=\"diffchange diffchange-inline\">-exemple-du-calcul-de-p|site=[[Encyclop\u00e6dia universalis]]|consult\u00e9 le=8 mars 2015}}.&lt;/ref&gt;.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== \u00c9tude syst\u00e9matique ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Le premier \u00e0 avoir syst\u00e9matis\u00e9 des algorithmes est le math\u00e9maticien [[Persans|perse]] </ins>[[<ins class=\"diffchange diffchange-inline\">Al-Khw\u00e2rizm\u00ee</ins>]]<ins class=\"diffchange diffchange-inline\">, actif entre 813 </ins>et <ins class=\"diffchange diffchange-inline\">833</ins>. <ins class=\"diffchange diffchange-inline\">Dans son ouvrage </ins>'<ins class=\"diffchange diffchange-inline\">'[[Abr\u00e9g\u00e9 du calcul par la restauration et </ins>la <ins class=\"diffchange diffchange-inline\">comparaison]]</ins>'', il <ins class=\"diffchange diffchange-inline\">\u00e9tudie toutes les [[\u00e9quation du second degr\u00e9|\u00e9quations du second degr\u00e9]] et en donne la r\u00e9solution par des algorithmes g\u00e9n\u00e9raux. Il </ins>utilise des <ins class=\"diffchange diffchange-inline\">m\u00e9thodes semblables \u00e0 celles </ins>des <ins class=\"diffchange diffchange-inline\">[[Math\u00e9matiques babyloniennes|Babyloniens]]</ins>, <ins class=\"diffchange diffchange-inline\">mais se diff\u00e9rencie par ses explications syst\u00e9matiques l\u00e0 o\u00f9 </ins>les <ins class=\"diffchange diffchange-inline\">Babyloniens donnaient seulement des exemples.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Le savant [[Al</ins>-<ins class=\"diffchange diffchange-inline\">Andalus|andalou]] </ins>[[<ins class=\"diffchange diffchange-inline\">Averro\u00e8s]] </ins>(<ins class=\"diffchange diffchange-inline\">[[1126]]-[[1198]]</ins>) <ins class=\"diffchange diffchange-inline\">\u00e9voque une m\u00e9thode de [[raisonnement</ins>]] <ins class=\"diffchange diffchange-inline\">o\u00f9 la th\u00e8se s\u2019affine \u00e9tape </ins>par <ins class=\"diffchange diffchange-inline\">\u00e9tape, it\u00e9rativement, jusqu\u2019\u00e0 une certaine convergence et ceci conform\u00e9ment au d\u00e9roulement d\u2019un algorithme. \u00c0 la m\u00eame \u00e9poque, au {{XIIe si\u00e8cle}}, le moine [[Adelard de Bath]] introduit le terme </ins>[[<ins class=\"diffchange diffchange-inline\">latin]] de {{lang|la|''algorismus''}}, par r\u00e9f\u00e9rence au nom de Al Khuwarizmi. Ce mot donne ''algorithme</ins>'' <ins class=\"diffchange diffchange-inline\">en fran\u00e7ais en [[1554</ins>]].</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Ce n'est qu'au </del>{{<del class=\"diffchange diffchange-inline\">s-|XIX</del>}} <del class=\"diffchange diffchange-inline\">que l'alg\u00e8bre lin\u00e9aire devient </del>une <del class=\"diffchange diffchange-inline\">branche des math\u00e9matiques </del>\u00e0 <del class=\"diffchange diffchange-inline\">part enti\u00e8re. </del>[[<del class=\"diffchange diffchange-inline\">Carl Friedrich Gauss</del>]] <del class=\"diffchange diffchange-inline\">trouve </del>[[<del class=\"diffchange diffchange-inline\">\u00c9limination </del>de <del class=\"diffchange diffchange-inline\">Gauss-Jordan|une </del>m\u00e9thode <del class=\"diffchange diffchange-inline\">g\u00e9n\u00e9rique</del>]] <del class=\"diffchange diffchange-inline\">pour la r\u00e9solution des syst\u00e8mes d'\u00e9quations lin\u00e9aires et [[Camille Jordan </del>(<del class=\"diffchange diffchange-inline\">math\u00e9maticien)|Camille Jordan]] r\u00e9sout d\u00e9finitivement le probl\u00e8me de la </del>[[<del class=\"diffchange diffchange-inline\">r\u00e9duction d'endomorphisme</del>]]<del class=\"diffchange diffchange-inline\">. En 1843</del>, <del class=\"diffchange diffchange-inline\">[[William Rowan Hamilton]] (inventeur du terme ''vector'') d\u00e9couvre les [[quaternion]]s ([[Alg\u00e8bre simple|extension]] </del>de <del class=\"diffchange diffchange-inline\">degr\u00e9 4 du [[corps commutatif</del>|<del class=\"diffchange diffchange-inline\">corps]] </del>des <del class=\"diffchange diffchange-inline\">[[nombre r\u00e9el|nombres r\u00e9els]])</del>. <del class=\"diffchange diffchange-inline\">En 1844</del>, [[<del class=\"diffchange diffchange-inline\">Hermann G\u00fcnther Grassmann</del>|<del class=\"diffchange diffchange-inline\">Hermann Grassmann</del>]] <del class=\"diffchange diffchange-inline\">publie son trait\u00e9 ''Die lineale Ausdehnungslehre'', ''La th\u00e9orie de l'extension lin\u00e9aire''</del>, <del class=\"diffchange diffchange-inline\">qui est la premi\u00e8re tentative de formalisation g\u00e9n\u00e9rale </del>de la <del class=\"diffchange diffchange-inline\">notion d'espace vectoriel. Si son \u0153uvre reste grandement inaper\u00e7ue, elle contient l'essentiel des id\u00e9es modernes de l'alg\u00e8bre lin\u00e9aire, et cette \u00e9tape fondamentale dans </del>le <del class=\"diffchange diffchange-inline\">d\u00e9veloppement </del>de <del class=\"diffchange diffchange-inline\">l'alg\u00e8bre lin\u00e9aire est reconnue comme telle tant par Hamilton que par </del>[[<del class=\"diffchange diffchange-inline\">Giuseppe Peano</del>]], qui <del class=\"diffchange diffchange-inline\">axiomatise enti\u00e8rement la th\u00e9orie </del>en <del class=\"diffchange diffchange-inline\">1888. Les espaces vectoriels deviennent alors une structure g\u00e9n\u00e9rale omnipr\u00e9sente dans presque tous les domaines math\u00e9matiques, notamment </del>en [[<del class=\"diffchange diffchange-inline\">analyse (math\u00e9matiques)|analyse]] ([[espace fonctionnel|espaces de fonctions</del>]]<del class=\"diffchange diffchange-inline\">)</del>.</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Au </ins>{{<ins class=\"diffchange diffchange-inline\">XVIIe si\u00e8cle</ins>}}<ins class=\"diffchange diffchange-inline\">, on pourrait entrevoir </ins>une <ins class=\"diffchange diffchange-inline\">certaine allusion </ins>\u00e0 <ins class=\"diffchange diffchange-inline\">la m\u00e9thode algorithmique chez </ins>[[<ins class=\"diffchange diffchange-inline\">Ren\u00e9 Descartes</ins>]] <ins class=\"diffchange diffchange-inline\">dans la m\u00e9thode g\u00e9n\u00e9rale propos\u00e9e par le </ins>[[<ins class=\"diffchange diffchange-inline\">Discours </ins>de <ins class=\"diffchange diffchange-inline\">la </ins>m\u00e9thode]] ([[<ins class=\"diffchange diffchange-inline\">1637</ins>]]<ins class=\"diffchange diffchange-inline\">), notamment quand, en sa deuxi\u00e8me partie</ins>, <ins class=\"diffchange diffchange-inline\">le math\u00e9maticien fran\u00e7ais propose </ins>de <ins class=\"diffchange diffchange-inline\">{{citation</ins>|<ins class=\"diffchange diffchange-inline\">diviser chacune </ins>des <ins class=\"diffchange diffchange-inline\">difficult\u00e9s que j\u2019examinerois, en autant de parcelles qu\u2019il se pourroit, et qu\u2019il seroit requis pour les mieux r\u00e9soudre}}</ins>. <ins class=\"diffchange diffchange-inline\">Sans \u00e9voquer explicitement les concepts de boucle</ins>, <ins class=\"diffchange diffchange-inline\">d\u2019it\u00e9ration ou de </ins>[[<ins class=\"diffchange diffchange-inline\">Recherche dichotomique</ins>|<ins class=\"diffchange diffchange-inline\">dichotomie</ins>]], <ins class=\"diffchange diffchange-inline\">l\u2019approche </ins>de <ins class=\"diffchange diffchange-inline\">Descartes pr\u00e9dispose </ins>la <ins class=\"diffchange diffchange-inline\">logique \u00e0 accueillir </ins>le <ins class=\"diffchange diffchange-inline\">concept </ins>de [[<ins class=\"diffchange diffchange-inline\">Programme informatique|programme</ins>]], <ins class=\"diffchange diffchange-inline\">mot </ins>qui <ins class=\"diffchange diffchange-inline\">na\u00eet </ins>en <ins class=\"diffchange diffchange-inline\">fran\u00e7ais </ins>en [[<ins class=\"diffchange diffchange-inline\">1677</ins>]].</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">== Int\u00e9r\u00eat ==</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">En 1843 , </ins>la <ins class=\"diffchange diffchange-inline\">math\u00e9maticienne et pionni\u00e8re des </ins>[[<ins class=\"diffchange diffchange-inline\">Informatique</ins>|<ins class=\"diffchange diffchange-inline\">sciences informatique</ins>]] [[<ins class=\"diffchange diffchange-inline\">Ada Lovelace</ins>]], <ins class=\"diffchange diffchange-inline\">fille de [</ins>[<ins class=\"diffchange diffchange-inline\">Lord Byron]] et assistante de </ins>[<ins class=\"diffchange diffchange-inline\">[Charles Babbage]] r\u00e9alise la premi\u00e8re [[Mise en \u0153uvre</ins>|<ins class=\"diffchange diffchange-inline\">impl\u00e9mentation</ins>]] <ins class=\"diffchange diffchange-inline\">d'un algorithme sous forme de programme (calcul des </ins>[[<ins class=\"diffchange diffchange-inline\">Nombre de Bernoulli</ins>|<ins class=\"diffchange diffchange-inline\">nombres de Bernoulli</ins>]]<ins class=\"diffchange diffchange-inline\">)&lt;ref&gt;</ins>[[<ins class=\"diffchange diffchange-inline\">Stephen Wolfram</ins>]] {{<ins class=\"diffchange diffchange-inline\">Lien web|langue=en|url =http://blog</ins>.<ins class=\"diffchange diffchange-inline\">stephenwolfram</ins>.<ins class=\"diffchange diffchange-inline\">com/2015/12/untangling-the-tale-of-ada</ins>-<ins class=\"diffchange diffchange-inline\">lovelace/|titre=Untangling the Tale of Ada Lovelace|site=blog.stephenwolfram</ins>.<ins class=\"diffchange diffchange-inline\">com</ins>}}<ins class=\"diffchange diffchange-inline\">&lt;/ref&gt;</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Sous leur forme </del>la <del class=\"diffchange diffchange-inline\">plus simple, les applications lin\u00e9aires dans les </del>[[<del class=\"diffchange diffchange-inline\">Espace vectoriel</del>|<del class=\"diffchange diffchange-inline\">espaces vectoriels</del>]] <del class=\"diffchange diffchange-inline\">repr\u00e9sentent intuitivement les d\u00e9placements dans les espaces g\u00e9om\u00e9triques \u00e9l\u00e9mentaires comme la </del>[[<del class=\"diffchange diffchange-inline\">Droite (math\u00e9matiques)|droite</del>]], <del class=\"diffchange diffchange-inline\">le </del>[[<del class=\"diffchange diffchange-inline\">Plan (math\u00e9matiques)</del>|<del class=\"diffchange diffchange-inline\">plan</del>]] <del class=\"diffchange diffchange-inline\">ou notre </del>[[<del class=\"diffchange diffchange-inline\">Espace (notion)#Physique</del>|<del class=\"diffchange diffchange-inline\">espace</del>]] <del class=\"diffchange diffchange-inline\">physique. Les bases de cette th\u00e9orie remplacent maintenant la repr\u00e9sentation construite par </del>[[<del class=\"diffchange diffchange-inline\">Euclide</del>]] <del class=\"diffchange diffchange-inline\">au </del>{{<del class=\"diffchange diffchange-inline\">IIIe si\u00e8cle av</del>. <del class=\"diffchange diffchange-inline\">J</del>.-<del class=\"diffchange diffchange-inline\">C</del>.}} <del class=\"diffchange diffchange-inline\">La construction moderne permet de g\u00e9n\u00e9raliser la notion d'espace \u00e0 des [[Dimension d'un espace vectoriel|dimensions]] quelconques</del>.</div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">L'alg\u00e8bre lin\u00e9aire permet de r\u00e9soudre tout un ensemble d'\u00e9quations dites lin\u00e9aires utilis\u00e9es non seulement en math\u00e9matiques ou en </del>[[<del class=\"diffchange diffchange-inline\">M\u00e9canique (science)|m\u00e9canique</del>]]<del class=\"diffchange diffchange-inline\">, mais aussi dans </del>de <del class=\"diffchange diffchange-inline\">nombreuses autres branches comme les </del>[[<del class=\"diffchange diffchange-inline\">Science </del>de <del class=\"diffchange diffchange-inline\">la nature</del>|<del class=\"diffchange diffchange-inline\">sciences naturelles</del>]] <del class=\"diffchange diffchange-inline\">ou les </del>[[<del class=\"diffchange diffchange-inline\">sciences sociales</del>]].</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Le </ins>[[<ins class=\"diffchange diffchange-inline\">dixi\u00e8me probl\u00e8me de Hilbert</ins>]] <ins class=\"diffchange diffchange-inline\">qui fait partie </ins>de <ins class=\"diffchange diffchange-inline\">la liste des {{nobr|23 </ins>[[<ins class=\"diffchange diffchange-inline\">Probl\u00e8mes </ins>de <ins class=\"diffchange diffchange-inline\">Hilbert</ins>|<ins class=\"diffchange diffchange-inline\">probl\u00e8mes</ins>]]<ins class=\"diffchange diffchange-inline\">}} pos\u00e9s par </ins>[[<ins class=\"diffchange diffchange-inline\">David Hilbert</ins>]] <ins class=\"diffchange diffchange-inline\">en 1900 \u00e0 Paris est clairement un probl\u00e8me algorithmique. En l'occurrence, la r\u00e9ponse est qu'il n'y a pas d'algorithme r\u00e9pondant au probl\u00e8me pos\u00e9</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Les espaces vectoriels forment aussi </del>un <del class=\"diffchange diffchange-inline\">outil fondamental </del>pour les [[<del class=\"diffchange diffchange-inline\">sciences de </del>l'<del class=\"diffchange diffchange-inline\">ing\u00e9nieur</del>]] <del class=\"diffchange diffchange-inline\">et servent de base \u00e0 de nombreux domaines dans la </del>[[<del class=\"diffchange diffchange-inline\">recherche op\u00e9rationnelle</del>]].</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== \u00c9poque contemporaine ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">L\u2019algorithmique des {{s2-|XX|e|XXI}} a pour fondement math\u00e9matique des formalismes, par exemple celui des [[machines de Turing]], qui permettent de d\u00e9finir pr\u00e9cis\u00e9ment ce qu'on entend par \u00ab \u00e9tapes \u00bb, par \u00ab pr\u00e9cis \u00bb et par \u00ab non ambigu \u00bb et qui donnent </ins>un <ins class=\"diffchange diffchange-inline\">cadre scientifique </ins>pour <ins class=\"diffchange diffchange-inline\">\u00e9tudier </ins>les <ins class=\"diffchange diffchange-inline\">propri\u00e9t\u00e9s des algorithmes. Cependant, suivant le formalisme choisi on obtient des approches algorithmiques diff\u00e9rentes pour r\u00e9soudre un m\u00eame probl\u00e8me. Par exemple l'</ins>[[<ins class=\"diffchange diffchange-inline\">Algorithme r\u00e9cursif|algorithmique r\u00e9cursive]], </ins>l'<ins class=\"diffchange diffchange-inline\">[[algorithme parall\u00e8le|algorithmique parall\u00e8le</ins>]] <ins class=\"diffchange diffchange-inline\">ou l\u2019</ins>[[<ins class=\"diffchange diffchange-inline\">informatique quantique</ins>]] <ins class=\"diffchange diffchange-inline\">donnent lieu \u00e0 des pr\u00e9sentations d'algorithmes diff\u00e9rentes de celles de l'algorithmique it\u00e9rative</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Enfin, c</del>'est <del class=\"diffchange diffchange-inline\">un outil utilis\u00e9 en math\u00e9matiques </del>dans <del class=\"diffchange diffchange-inline\">des domaines aussi divers que </del>la <del class=\"diffchange diffchange-inline\">[[th\u00e9orie </del>des <del class=\"diffchange diffchange-inline\">groupes]]</del>, [[<del class=\"diffchange diffchange-inline\">Th\u00e9orie des anneaux|des anneaux</del>]] <del class=\"diffchange diffchange-inline\">ou </del>[[<del class=\"diffchange diffchange-inline\">Corps commutatif|des corps</del>]], <del class=\"diffchange diffchange-inline\">l</del>'<del class=\"diffchange diffchange-inline\">[[Analyse fonctionnelle (</del>math\u00e9matiques<del class=\"diffchange diffchange-inline\">)|</del>analyse <del class=\"diffchange diffchange-inline\">fonctionnelle]], la [[g\u00e9om\u00e9trie diff\u00e9rentielle]] ou la [[th\u00e9orie des nombres]]</del>.</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">L'algorithmique s</ins>'est <ins class=\"diffchange diffchange-inline\">surtout d\u00e9velopp\u00e9e </ins>dans la <ins class=\"diffchange diffchange-inline\">deuxi\u00e8me moiti\u00e9 du {{s-|XX}}, comme support conceptuel de la programmation </ins>des <ins class=\"diffchange diffchange-inline\">ordinateurs</ins>, <ins class=\"diffchange diffchange-inline\">dans le cadre du d\u00e9veloppement de l'informatique pendant cette p\u00e9riode. </ins>[[<ins class=\"diffchange diffchange-inline\">Donald Knuth</ins>]]<ins class=\"diffchange diffchange-inline\">, auteur du trait\u00e9 ''</ins>[[<ins class=\"diffchange diffchange-inline\">The Art of Computer Programming</ins>]]<ins class=\"diffchange diffchange-inline\">'' qui d\u00e9crit de tr\u00e8s nombreux algorithmes, a contribu\u00e9</ins>, <ins class=\"diffchange diffchange-inline\">avec d</ins>'<ins class=\"diffchange diffchange-inline\">autres, \u00e0 poser les fondements </ins>math\u00e9matiques <ins class=\"diffchange diffchange-inline\">de leur </ins>analyse.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>== <del class=\"diffchange diffchange-inline\">Pr\u00e9sentation \u00e9l\u00e9mentaire </del>==</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>== <ins class=\"diffchange diffchange-inline\">Vocabulaire </ins>==</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">L</del>'<del class=\"diffchange diffchange-inline\">alg\u00e8bre lin\u00e9aire commence par </del>l'<del class=\"diffchange diffchange-inline\">\u00e9tude </del>de <del class=\"diffchange diffchange-inline\">[[vecteur]]s dans les espaces cart\u00e9siens de dimension 2 et 3</del>. <del class=\"diffchange diffchange-inline\">Un vecteur, ici, </del>est <del class=\"diffchange diffchange-inline\">une [[Relation d'\u00e9quivalence|classe d'\u00e9quivalence]] de bipoints qui unifie les segments de droite caract\u00e9ris\u00e9s \u00e0 la fois par leur longueur (ou ''norme''), leur direction et leur sens : deux bipoints repr\u00e9sentent un m\u00eame vecteur si le quadrilat\u00e8re form\u00e9 sur les quatre points est un [[parall\u00e9logramme]]. Les vecteurs peuvent alors \u00eatre utilis\u00e9s pour repr\u00e9senter certaines entit\u00e9s physiques </del>comme <del class=\"diffchange diffchange-inline\">des d\u00e9placements, additionn\u00e9s entre eux ou encore multipli\u00e9s par des scalaires (''nombres''), formant ainsi le premier exemple concret d'espace vectoriel</del>.</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Le substantif </ins>'<ins class=\"diffchange diffchange-inline\">'algorithmique'' d\u00e9signe </ins>l'<ins class=\"diffchange diffchange-inline\">ensemble des m\u00e9thodes permettant </ins>de <ins class=\"diffchange diffchange-inline\">cr\u00e9er des algorithmes</ins>. <ins class=\"diffchange diffchange-inline\">Le terme </ins>est <ins class=\"diffchange diffchange-inline\">\u00e9galement employ\u00e9 </ins>comme <ins class=\"diffchange diffchange-inline\">adjectif</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">L</del>'<del class=\"diffchange diffchange-inline\">alg\u00e8bre lin\u00e9aire moderne s</del>'<del class=\"diffchange diffchange-inline\">int\u00e9resse beaucoup aux espaces de [[Dimension d</del>'un <del class=\"diffchange diffchange-inline\">espace vectoriel|dimension arbitraire, \u00e9ventuellement infinie]]. La plupart des r\u00e9sultats obtenus en dimension 2 ou 3 peuvent \u00eatre \u00e9tendus aux dimensions finies sup\u00e9rieures, ce qui permet une interpr\u00e9tation g\u00e9om\u00e9trique de listes de nombres (une liste de </del>''<del class=\"diffchange diffchange-inline\">n'' nombres s'interpr\u00e9tant comme un vecteur d'un espace </del>\u00e0 ''<del class=\"diffchange diffchange-inline\">n'' dimensions)</del>.</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Un </ins>''<ins class=\"diffchange diffchange-inline\">algorithme</ins>'<ins class=\"diffchange diffchange-inline\">' \u00e9nonce une solution \u00e0 </ins>un <ins class=\"diffchange diffchange-inline\">probl\u00e8me sous la forme d\u2019un encha\u00eenement d\u2019</ins>''<ins class=\"diffchange diffchange-inline\">op\u00e9rations </ins>\u00e0 <ins class=\"diffchange diffchange-inline\">effectuer</ins>''.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">== Quelques th\u00e9or\u00e8mes ==</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Les informaticiens utilisent fr\u00e9quemment l\u2019anglicisme </ins>''<ins class=\"diffchange diffchange-inline\">impl\u00e9mentation</ins>'' <ins class=\"diffchange diffchange-inline\">pour d\u00e9signer la mise en \u0153uvre de </ins>l'<ins class=\"diffchange diffchange-inline\">algorithme </ins>dans un [[<ins class=\"diffchange diffchange-inline\">langage de programmation</ins>]]<ins class=\"diffchange diffchange-inline\">. Cette impl\u00e9mentation r\u00e9alise la transcription des op\u00e9rations constitutives </ins>de <ins class=\"diffchange diffchange-inline\">l\u2019algorithme </ins>et <ins class=\"diffchange diffchange-inline\">pr\u00e9cise la fa\u00e7on dont ces op\u00e9rations sont invoqu\u00e9es. Cette \u00e9criture en </ins>[[<ins class=\"diffchange diffchange-inline\">langage informatique</ins>]]<ins class=\"diffchange diffchange-inline\">, est aussi fr\u00e9quemment d\u00e9sign\u00e9e par le terme </ins>de <ins class=\"diffchange diffchange-inline\">\u00ab </ins>''[[<ins class=\"diffchange diffchange-inline\">codage </ins>(<ins class=\"diffchange diffchange-inline\">programmation</ins>)|<ins class=\"diffchange diffchange-inline\">codage</ins>]]'' <ins class=\"diffchange diffchange-inline\">\u00bb&lt;ref&gt;</ins>En [[<ins class=\"diffchange diffchange-inline\">cryptographie</ins>]], <ins class=\"diffchange diffchange-inline\">le terme codage </ins>est <ins class=\"diffchange diffchange-inline\">utilis\u00e9 dans un sens diff\u00e9rent</ins>.&lt;/<ins class=\"diffchange diffchange-inline\">ref</ins>&gt;<ins class=\"diffchange diffchange-inline\">. On parle de ''\u00ab </ins>[[<ins class=\"diffchange diffchange-inline\">code source</ins>]] <ins class=\"diffchange diffchange-inline\">\u00bb</ins>'<ins class=\"diffchange diffchange-inline\">' pour d\u00e9signer le texte, constituant le programme, r\u00e9alisant l\u2019algorithme</ins>. <ins class=\"diffchange diffchange-inline\">Le ''code'' est plus ou moins d\u00e9taill\u00e9 selon le niveau d\u2019abstraction du langage utilis\u00e9, </ins>de <ins class=\"diffchange diffchange-inline\">m\u00eame qu</ins>'<ins class=\"diffchange diffchange-inline\">une recette </ins>de <ins class=\"diffchange diffchange-inline\">cuisine doit \u00eatre plus ou moins d\u00e9taill\u00e9e selon l\u2019exp\u00e9rience du cuisinier</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* [[Th\u00e9or\u00e8me de la base incompl\u00e8te]]&lt;ref&gt;Dans le cas o\u00f9 </del>''<del class=\"diffchange diffchange-inline\">G </del>''<del class=\"diffchange diffchange-inline\">est infinie, ce th\u00e9or\u00e8me utilise </del>l'<del class=\"diffchange diffchange-inline\">[[axiome du choix]], qui intervient de m\u00eame </del>dans <del class=\"diffchange diffchange-inline\">les \u00e9nonc\u00e9s suivants en dimension infinie.&lt;/ref&gt; : soient ''E ''</del>un <del class=\"diffchange diffchange-inline\">espace vectoriel, ''G ''une </del>[[<del class=\"diffchange diffchange-inline\">famille g\u00e9n\u00e9ratrice</del>]] de <del class=\"diffchange diffchange-inline\">''E ''</del>et <del class=\"diffchange diffchange-inline\">''L ''une </del>[[<del class=\"diffchange diffchange-inline\">Ind\u00e9pendance lin\u00e9aire|famille libre</del>]] de <del class=\"diffchange diffchange-inline\">vecteurs de ''E</del>''<del class=\"diffchange diffchange-inline\">. Alors il existe au moins une </del>[[<del class=\"diffchange diffchange-inline\">base </del>(<del class=\"diffchange diffchange-inline\">alg\u00e8bre lin\u00e9aire</del>)|<del class=\"diffchange diffchange-inline\">base</del>]] <del class=\"diffchange diffchange-inline\">de </del>''<del class=\"diffchange diffchange-inline\">E ''form\u00e9e en prenant la r\u00e9union de ''L ''et d'une partie de ''G''.</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* </del>En <del class=\"diffchange diffchange-inline\">particulier, tout espace vectoriel poss\u00e8de au moins une base.</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* Toutes les bases d'un m\u00eame espace vectoriel ont le m\u00eame </del>[[<del class=\"diffchange diffchange-inline\">Cardinalit\u00e9 (math\u00e9matiques)|cardinal]].</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* Tout espace vectoriel A poss\u00e8de un [[espace dual]] A* ; si A est [[Espace vectoriel de dimension finie|de dimension finie</del>]], <del class=\"diffchange diffchange-inline\">A* </del>est <del class=\"diffchange diffchange-inline\">de m\u00eame dimension</del>.</div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* [[Formule de Grassmann]] : Soient &lt;math&gt;F</del>&lt;/<del class=\"diffchange diffchange-inline\">math</del>&gt; <del class=\"diffchange diffchange-inline\">et &lt;math&gt;G&lt;/math&gt; deux </del>[[<del class=\"diffchange diffchange-inline\">Sous-espace vectoriel|sous-espaces vectoriels</del>]] <del class=\"diffchange diffchange-inline\">d</del>'<del class=\"diffchange diffchange-inline\">un m\u00eame espace vectoriel</del>. <del class=\"diffchange diffchange-inline\">On a alors :&lt;!--MERCI DE NE PAS MODIFIER LES SIGNES DE CETTE FORMULE : en cas </del>de <del class=\"diffchange diffchange-inline\">doute, consulter l</del>'<del class=\"diffchange diffchange-inline\">article \"Formule </del>de <del class=\"diffchange diffchange-inline\">Grassmann\"--&gt;&lt;center&gt;&lt;math&gt;\\dim(F)+\\dim(G)=\\dim(F+G)+\\dim(F\\cap G)</del>.<del class=\"diffchange diffchange-inline\">&lt;/math&gt;&lt;/center&gt;</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">D'autres th\u00e9or\u00e8mes concernent </del>les <del class=\"diffchange diffchange-inline\">conditions d'inversion de [[Matrice (math\u00e9matiques)|matrices]] de divers types </del>:</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== \u00c9tude formelle ==</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* <del class=\"diffchange diffchange-inline\">[[matrice diagonale]] </del>;</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">De nombreux outils formels ou th\u00e9oriques ont \u00e9t\u00e9 d\u00e9velopp\u00e9s pour d\u00e9crire </ins>les <ins class=\"diffchange diffchange-inline\">algorithmes, les \u00e9tudier, exprimer leurs qualit\u00e9s, pouvoir les comparer </ins>:</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">&lt;!--* bande ?--&gt;</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>* <ins class=\"diffchange diffchange-inline\">ainsi, pour d\u00e9crire les algorithmes, des structures algorithmiques ont \u00e9t\u00e9 mises en \u00e9vidence : structures de contr\u00f4le et structures de donn\u00e9es </ins>;</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* [[<del class=\"diffchange diffchange-inline\">matrice triangulaire</del>]] ;</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>* <ins class=\"diffchange diffchange-inline\">pour justifier de la qualit\u00e9 des algorithmes, les notions de correction, de </ins>[[<ins class=\"diffchange diffchange-inline\">Complet (complexit\u00e9)|compl\u00e9tude</ins>]] <ins class=\"diffchange diffchange-inline\">et de terminaison ont \u00e9t\u00e9 mises en place </ins>;</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* [[<del class=\"diffchange diffchange-inline\">matrice \u00e0 diagonale dominante</del>]].</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>* <ins class=\"diffchange diffchange-inline\">enfin, pour comparer les algorithmes, une </ins>[[<ins class=\"diffchange diffchange-inline\">Th\u00e9orie de la complexit\u00e9 (informatique th\u00e9orique)|th\u00e9orie de la complexit\u00e9</ins>]] <ins class=\"diffchange diffchange-inline\">des algorithmes a \u00e9t\u00e9 d\u00e9finie</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Un th\u00e9or\u00e8me int\u00e9ressant \u00e0 </del>l'<del class=\"diffchange diffchange-inline\">\u00e9poque des m\u00e9moires d'ordinateurs </del>de <del class=\"diffchange diffchange-inline\">petite taille \u00e9tait qu'on pouvait travailler s\u00e9par\u00e9ment sur des sous-ensembles (\u00ab blocs \u00bb) d'une matrice en </del>les <del class=\"diffchange diffchange-inline\">combinant ensuite par </del>les <del class=\"diffchange diffchange-inline\">m\u00eames r\u00e8gles qu'on utilise pour combiner des scalaires dans les matrices </del>(<del class=\"diffchange diffchange-inline\">''cf''. l\u2019article </del>[[<del class=\"diffchange diffchange-inline\">Matrice par blocs</del>]]). <del class=\"diffchange diffchange-inline\">Avec </del>les <del class=\"diffchange diffchange-inline\">m\u00e9moires actuelles de plusieurs </del>[[<del class=\"diffchange diffchange-inline\">Octet</del>|<del class=\"diffchange diffchange-inline\">gigaoctets</del>]], <del class=\"diffchange diffchange-inline\">cette question a perdu un peu de son int\u00e9r\u00eat pratique</del>, <del class=\"diffchange diffchange-inline\">mais reste tr\u00e8s pris\u00e9e en </del>[[<del class=\"diffchange diffchange-inline\">th\u00e9orie des nombres</del>]]<del class=\"diffchange diffchange-inline\">, pour la </del>[[<del class=\"diffchange diffchange-inline\">d\u00e9composition en produit de facteurs premiers</del>]] <del class=\"diffchange diffchange-inline\">avec le </del>[[<del class=\"diffchange diffchange-inline\">Crible alg\u00e9brique</del>|<del class=\"diffchange diffchange-inline\">crible g\u00e9n\u00e9ral de corps de nombres</del>]] <del class=\"diffchange diffchange-inline\">(''</del>[[<del class=\"diffchange diffchange-inline\">algorithme </del>de <del class=\"diffchange diffchange-inline\">Lanczos</del>]]<del class=\"diffchange diffchange-inline\">''</del>).</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== Structures algorithmiques ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Les concepts en \u0153uvre en algorithmique, par exemple selon </ins>l'<ins class=\"diffchange diffchange-inline\">approche </ins>de <ins class=\"diffchange diffchange-inline\">[[Niklaus Wirth|N. Wirth]] pour </ins>les <ins class=\"diffchange diffchange-inline\">langages </ins>les <ins class=\"diffchange diffchange-inline\">plus r\u00e9pandus ([[Pascal </ins>(<ins class=\"diffchange diffchange-inline\">langage)|Pascal]], </ins>[[<ins class=\"diffchange diffchange-inline\">C (langage)|C</ins>]]<ins class=\"diffchange diffchange-inline\">{{etc.}}</ins>)<ins class=\"diffchange diffchange-inline\">, sont en petit nombre</ins>. <ins class=\"diffchange diffchange-inline\">Ils appartiennent \u00e0 deux classes :</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* </ins>les [[<ins class=\"diffchange diffchange-inline\">structure de contr\u00f4le</ins>|<ins class=\"diffchange diffchange-inline\">structures de contr\u00f4le</ins>]] <ins class=\"diffchange diffchange-inline\">:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** s\u00e9quences</ins>,</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** conditionnelles</ins>,</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** boucles ;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* les </ins>[[<ins class=\"diffchange diffchange-inline\">Structure de donn\u00e9es|structures de donn\u00e9es</ins>]] <ins class=\"diffchange diffchange-inline\">:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** </ins>[[<ins class=\"diffchange diffchange-inline\">Constante|constantes</ins>]]<ins class=\"diffchange diffchange-inline\">,</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** </ins>[[<ins class=\"diffchange diffchange-inline\">Variable (informatique)</ins>|<ins class=\"diffchange diffchange-inline\">variables</ins>]]<ins class=\"diffchange diffchange-inline\">,</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** </ins>[[<ins class=\"diffchange diffchange-inline\">Tableau (structure </ins>de <ins class=\"diffchange diffchange-inline\">donn\u00e9es)|tableaux</ins>]] <ins class=\"diffchange diffchange-inline\">;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** structures r\u00e9cursives (listes, arbres, graphes</ins>).</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">== Utilisations ==</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Ce d\u00e9coupage </ins>est <ins class=\"diffchange diffchange-inline\">parfois difficile \u00e0 percevoir </ins>pour <ins class=\"diffchange diffchange-inline\">certains langages ([[Lisp (langage)|Lisp]]</ins>, [[<ins class=\"diffchange diffchange-inline\">Prolog</ins>]]<ins class=\"diffchange diffchange-inline\">\u2026) plus bas\u00e9s sur la notion </ins>de [[<ins class=\"diffchange diffchange-inline\">algorithme r\u00e9cursif</ins>|<ins class=\"diffchange diffchange-inline\">r\u00e9cursivit\u00e9</ins>]] <ins class=\"diffchange diffchange-inline\">o\u00f9 certaines structures de contr\u00f4le sont implicites et, donc, semblent dispara\u00eetre</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Les espaces vectoriels forment le support et le fondement de l'alg\u00e8bre lin\u00e9aire. Ils sont aussi pr\u00e9sents dans de nombreux domaines distincts. S'il n'</del>est <del class=\"diffchange diffchange-inline\">pas possible d'indiquer ici tous les cas d'utilisation, on peut tout de m\u00eame citer </del>pour <del class=\"diffchange diffchange-inline\">les principales structures objet de th\u00e9ories, des exemples significatifs. Leurs r\u00f4les dans de vastes th\u00e9ories ne traitant pas d'une structure particuli\u00e8re</del>, <del class=\"diffchange diffchange-inline\">comme celles des </del>[[<del class=\"diffchange diffchange-inline\">th\u00e9orie alg\u00e9brique des nombres|nombres alg\u00e9briques</del>]] <del class=\"diffchange diffchange-inline\">ou\u00a0 </del>de [[<del class=\"diffchange diffchange-inline\">Th\u00e9orie de Galois</del>|<del class=\"diffchange diffchange-inline\">Galois</del>]] <del class=\"diffchange diffchange-inline\">peuvent aussi \u00eatre \u00e9voqu\u00e9s</del>.</div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Les espaces vectoriels utilis\u00e9s sont d'une grande diversit\u00e9. On y trouve les classiques espaces vectoriels de dimension 2 ou 3 sur les [[nombre r\u00e9el|nombres r\u00e9els]]</del>, <del class=\"diffchange diffchange-inline\">cependant la dimension peut \u00eatre quelconque</del>, <del class=\"diffchange diffchange-inline\">m\u00eame infinie. Les nombres complexes </del>sont <del class=\"diffchange diffchange-inline\">aussi tr\u00e8s utilis\u00e9s</del>, <del class=\"diffchange diffchange-inline\">ainsi que les [[nombre rationnel|rationnels]]. Il n'est pas rare </del>qu'<del class=\"diffchange diffchange-inline\">une partie des nombres r\u00e9els ou complexes soit consid\u00e9r\u00e9 comme </del>un <del class=\"diffchange diffchange-inline\">espace vectoriel rationnel. Le corps de base peut aussi contenir un nombre fini d'\u00e9l\u00e9ments, d\u00e9finissant parfois </del>un <del class=\"diffchange diffchange-inline\">[[espace vectoriel fini]]</del>.</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== Correction, compl\u00e9tude, terminaison ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Ces trois notions \u00ab correction \u00bb</ins>, <ins class=\"diffchange diffchange-inline\">\u00ab compl\u00e9tude \u00bb</ins>, <ins class=\"diffchange diffchange-inline\">\u00ab terminaison \u00bb </ins>sont <ins class=\"diffchange diffchange-inline\">li\u00e9es</ins>, <ins class=\"diffchange diffchange-inline\">et supposent </ins>qu'un <ins class=\"diffchange diffchange-inline\">algorithme est \u00e9crit pour r\u00e9soudre </ins>un <ins class=\"diffchange diffchange-inline\">probl\u00e8me</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Les propri\u00e9t\u00e9s g\u00e9om\u00e9triques de la structure permettent la d\u00e9monstration de nombreux th\u00e9or\u00e8mes. Elles ne se limitent pas aux cas o\u00f9 l'espace est r\u00e9el, m\u00eame dans le cas de corps plus insolites comme les </del>[[<del class=\"diffchange diffchange-inline\">corps fini]]s ou les [[extension finie</del>|<del class=\"diffchange diffchange-inline\">extensions finies</del>]] <del class=\"diffchange diffchange-inline\">des rationnels, les propri\u00e9t\u00e9s g\u00e9om\u00e9triques s</del>'<del class=\"diffchange diffchange-inline\">av\u00e8rent parfois essentielles</del>.</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">La </ins>[[<ins class=\"diffchange diffchange-inline\">Terminaison d'un algorithme</ins>|<ins class=\"diffchange diffchange-inline\">terminaison</ins>]] <ins class=\"diffchange diffchange-inline\">est l'assurance que l'algorithme se\u00a0 terminera en un temps fini. Les preuves le plus simples de terminaison font intervenir une fonction \u00e0 valeurs enti\u00e8res positives strictement d\u00e9croissante \u00e0 chaque \u00ab pas \u00bb de l</ins>'<ins class=\"diffchange diffchange-inline\">algorithme</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">=== Groupe fini ===</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\u00c9tant donn\u00e9 la garantie qu</ins>'un <ins class=\"diffchange diffchange-inline\">algorithme se terminera, la preuve de correction doit apporter l'assurance que si l'algorithme se termine en donnant un r\u00e9sultat, alors ce r\u00e9sultat est effectivement une solution au probl\u00e8me pos\u00e9. Les preuves de correction font intervenir une sp\u00e9cification logique que doivent v\u00e9rifier les solutions </ins>du <ins class=\"diffchange diffchange-inline\">probl\u00e8me</ins>. <ins class=\"diffchange diffchange-inline\">La preuve de correction consiste donc \u00e0 montrer que les r\u00e9sultats de l</ins>'<ins class=\"diffchange diffchange-inline\">algorithme satisfait cette sp\u00e9cification</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">{{Article d\u00e9taill\u00e9|Repr\u00e9sentations d</del>'un <del class=\"diffchange diffchange-inline\">groupe fini}}</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">[[Fichier:Rotations </del>du <del class=\"diffchange diffchange-inline\">cube</del>.<del class=\"diffchange diffchange-inline\">jpg|thumb|[[Repr\u00e9sentations du groupe sym\u00e9trique|Repr\u00e9sentation du groupe sym\u00e9trique d</del>'<del class=\"diffchange diffchange-inline\">indice 4]] comme groupe des rotations du cube dans un espace vectoriel de dimension 3</del>.<del class=\"diffchange diffchange-inline\">]]</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>La <del class=\"diffchange diffchange-inline\">[[Groupe fini#Classification des groupes finis|classification des groupes finis]] est une vaste question</del>, <del class=\"diffchange diffchange-inline\">encore objet </del>de <del class=\"diffchange diffchange-inline\">recherche. Si le groupe contient un petit nombre d</del>'<del class=\"diffchange diffchange-inline\">\u00e9l\u00e9ments</del>, <del class=\"diffchange diffchange-inline\">les [[th\u00e9or\u00e8mes </del>de <del class=\"diffchange diffchange-inline\">Sylow]] peuvent suffire </del>pour <del class=\"diffchange diffchange-inline\">en d\u00e9terminer la structure. Une m\u00e9thode beaucoup plus puissante est n\u00e9cessaire dans </del>le <del class=\"diffchange diffchange-inline\">cas g\u00e9n\u00e9ral</del>.</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>La <ins class=\"diffchange diffchange-inline\">preuve de compl\u00e9tude garantit que</ins>, <ins class=\"diffchange diffchange-inline\">pour un espace </ins>de <ins class=\"diffchange diffchange-inline\">probl\u00e8mes donn\u00e9, l'algorithme, s</ins>'<ins class=\"diffchange diffchange-inline\">il se termine</ins>, <ins class=\"diffchange diffchange-inline\">donnera l'ensemble des solutions de l'espace du probl\u00e8me. Les preuves </ins>de <ins class=\"diffchange diffchange-inline\">compl\u00e9tude demandent \u00e0 identifier l'espace du probl\u00e8me et l'espace des solutions </ins>pour <ins class=\"diffchange diffchange-inline\">ensuite montrer que l'algorithme produit bien </ins>le <ins class=\"diffchange diffchange-inline\">second \u00e0 partir du premier</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">[[Ferdinand Georg Frobenius|Georg Frobenius]], \u00e0 la suite de travaux de [[Richard Dedekind]], d\u00e9veloppe une nouvelle th\u00e9orie&lt;ref&gt;{{en}} [[Charles W. Curtis|C. W. Curtis]], \u00ab{{lang|en|texte</del>=<del class=\"diffchange diffchange-inline\">\u00a0Representation theory of finite groups, from Frobenius to Brauer\u00a0}}\u00bb, dans ''[[The Mathematical Intelligencer|Math. Intelligencer]]'', 1992, </del>{{<del class=\"diffchange diffchange-inline\">p.</del>|<del class=\"diffchange diffchange-inline\">48-57</del>}}<del class=\"diffchange diffchange-inline\">&lt;/ref&gt; en [[1896 en science|1896]]. Elle se fonde sur l'id\u00e9e que l'ensemble des \u00ab [[sym\u00e9trie]]s \u00bb (au sens : [[Application lin\u00e9aire#Cas particuliers|automorphismes]]) d'un espace vectoriel poss\u00e8de une structure de groupe. Il est toujours possible de ''repr\u00e9senter'' un groupe fini par des \u00ab sym\u00e9tries \u00bb bien choisies sur un espace vectoriel de dimension suffisante. Un groupe est ainsi incarn\u00e9 par des [[Transformation g\u00e9om\u00e9trique|transformations g\u00e9om\u00e9triques]] simples. Une telle incarnation prend le nom de ''repr\u00e9sentation d'un groupe''.</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>=<ins class=\"diffchange diffchange-inline\">== Complexit\u00e9 algorithmique ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>{{<ins class=\"diffchange diffchange-inline\">Article d\u00e9taill\u00e9</ins>|<ins class=\"diffchange diffchange-inline\">Th\u00e9orie de la complexit\u00e9 des algorithmes</ins>}}</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>Les <del class=\"diffchange diffchange-inline\">espaces vectoriels choisis </del>sont <del class=\"diffchange diffchange-inline\">de dimension finie, en g\u00e9n\u00e9ral sur le corps des complexes&lt;ref&gt;Les 11 premiers chapitres de {{Serre2}} ne concernent que </del>les <del class=\"diffchange diffchange-inline\">espaces vectoriels complexes.&lt;/ref&gt;, cependant pour disposer de bonnes propri\u00e9t\u00e9s arithm\u00e9tiques le corps peut \u00eatre celui des </del>[[<del class=\"diffchange diffchange-inline\">nombre rationnel</del>|<del class=\"diffchange diffchange-inline\">rationnels</del>]]<del class=\"diffchange diffchange-inline\">&lt;ref&gt;{{en}} [[Walter Feit]], </del>''<del class=\"diffchange diffchange-inline\">Characters of finite groups</del>'', <del class=\"diffchange diffchange-inline\">Benjamin</del>, <del class=\"diffchange diffchange-inline\">1967&lt;/ref&gt; ou encore utiliser des </del>[[<del class=\"diffchange diffchange-inline\">entier alg\u00e9brique|entiers alg\u00e9briques</del>]] <del class=\"diffchange diffchange-inline\">comme pour la d\u00e9monstration du [[Th\u00e9or\u00e8me </del>de <del class=\"diffchange diffchange-inline\">Burnside (groupe r\u00e9soluble)|th\u00e9or\u00e8me de Burnside sur les groupes r\u00e9solubles]]&lt;ref&gt;{{en}} [[William Burnside]], </del>''<del class=\"diffchange diffchange-inline\">Theory of Groups of Finite Order</del>'', <del class=\"diffchange diffchange-inline\">Dover, 2004&lt;/ref&gt;. </del>[[<del class=\"diffchange diffchange-inline\">Richard Brauer</del>]] <del class=\"diffchange diffchange-inline\">\u00e9tudie un cas tr\u00e8s abstrait</del>, <del class=\"diffchange diffchange-inline\">celui des repr\u00e9sentations sur un espace vectoriel construit \u00e0 l'aide d'un [[corps fini]]&lt;ref&gt;</del>{{<del class=\"diffchange diffchange-inline\">de</del>}} <del class=\"diffchange diffchange-inline\">[[Richard Brauer]], \u00ab \u00dcber die Darstellung von Gruppen in Galoisschen Feldern \u00bb, </del>dans <del class=\"diffchange diffchange-inline\">''[[Hermann (\u00e9ditions)#1876-1956|Act. Sci. Ind.]]'', vol. 195, 1935&lt;/ref&gt;</del>.</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>Les <ins class=\"diffchange diffchange-inline\">principales notions math\u00e9matiques dans le calcul du co\u00fbt d\u2019un algorithme pr\u00e9cis </ins>sont les [[<ins class=\"diffchange diffchange-inline\">notation de Landau</ins>|<ins class=\"diffchange diffchange-inline\">notions de domination</ins>]] <ins class=\"diffchange diffchange-inline\">(not\u00e9e </ins>''<ins class=\"diffchange diffchange-inline\">O(f(n))</ins>'', <ins class=\"diffchange diffchange-inline\">\u00ab grand o \u00bb)</ins>, <ins class=\"diffchange diffchange-inline\">o\u00f9 ''f'' est une </ins>[[<ins class=\"diffchange diffchange-inline\">fonction math\u00e9matique</ins>]] de ''<ins class=\"diffchange diffchange-inline\">n</ins>'', <ins class=\"diffchange diffchange-inline\">variable d\u00e9signant la quantit\u00e9 d\u2019informations (en </ins>[[<ins class=\"diffchange diffchange-inline\">bit</ins>]]<ins class=\"diffchange diffchange-inline\">s</ins>, <ins class=\"diffchange diffchange-inline\">en nombre d\u2019enregistrements</ins>{{<ins class=\"diffchange diffchange-inline\">etc.</ins>}}<ins class=\"diffchange diffchange-inline\">) manipul\u00e9e </ins>dans <ins class=\"diffchange diffchange-inline\">l\u2019algorithme</ins>. <ins class=\"diffchange diffchange-inline\">En algorithmique on trouve souvent des complexit\u00e9s du type :</ins></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Un exemple relativement simple d'utilisation </del>de <del class=\"diffchange diffchange-inline\">cette th\u00e9orie est donn\u00e9 par Burnside, avec [[Th\u00e9or\u00e8me </del>de <del class=\"diffchange diffchange-inline\">Burnside (probl\u00e8me </del>de <del class=\"diffchange diffchange-inline\">1902</del>)|<del class=\"diffchange diffchange-inline\">son th\u00e9or\u00e8me]] sur les [[sous</del>-<del class=\"diffchange diffchange-inline\">groupe]]s d'[[Exposant d'un groupe</del>|<del class=\"diffchange diffchange-inline\">exposant]] fini du [[groupe </del>(<del class=\"diffchange diffchange-inline\">math\u00e9matiques</del>)|<del class=\"diffchange diffchange-inline\">groupe]] [[Groupe g\u00e9n\u00e9ral </del>lin\u00e9aire|lin\u00e9aire<del class=\"diffchange diffchange-inline\">]] GL</del>(<del class=\"diffchange diffchange-inline\">''</del>n<del class=\"diffchange diffchange-inline\">'', [[nombre complexe</del>|<del class=\"diffchange diffchange-inline\">\u2102]]</del>)<del class=\"diffchange diffchange-inline\">.</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{|class=\"wikitable\"</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">! Notation</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">! Type </ins>de <ins class=\"diffchange diffchange-inline\">complexit\u00e9</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|-</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|&lt;math&gt;O(1)&lt;/math&gt;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|complexit\u00e9 constante (ind\u00e9pendante </ins>de <ins class=\"diffchange diffchange-inline\">la taille </ins>de <ins class=\"diffchange diffchange-inline\">la donn\u00e9e</ins>)</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>|-</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>|<ins class=\"diffchange diffchange-inline\">&lt;math&gt;O(\\log</ins>(<ins class=\"diffchange diffchange-inline\">n</ins>)<ins class=\"diffchange diffchange-inline\">)&lt;/math&gt;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>|<ins class=\"diffchange diffchange-inline\">complexit\u00e9 logarithmique</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|-</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|&lt;math&gt;O(n)&lt;/math&gt;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|complexit\u00e9 </ins>lin\u00e9aire</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>|<ins class=\"diffchange diffchange-inline\">-</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|&lt;math&gt;O(n \\log(n))&lt;/math&gt;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|complexit\u00e9 quasi </ins>lin\u00e9aire</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|-</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|&lt;math&gt;O(n^{2})&lt;/math&gt;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|complexit\u00e9 quadratique</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|-</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|&lt;math&gt;O(n^{3})&lt;/math&gt;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|complexit\u00e9 cubique</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|-</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|&lt;math&gt;O</ins>(n<ins class=\"diffchange diffchange-inline\">^p)&lt;/math&gt;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>|<ins class=\"diffchange diffchange-inline\">complexit\u00e9 polynomiale</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|-</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|&lt;math&gt;O(n^{\\log(n</ins>)<ins class=\"diffchange diffchange-inline\">})&lt;/math&gt;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|complexit\u00e9 quasi polynomiale</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|-</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|&lt;math&gt;O(2^{n})&lt;/math&gt;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|complexit\u00e9 exponentielle</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|-</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|&lt;math&gt;O(n!)&lt;/math&gt;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|complexit\u00e9 factorielle</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|}</ins></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">=== Anneau ===</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Sans entrer dans les d\u00e9tails math\u00e9matiques, le calcul de l\u2019efficacit\u00e9 d\u2019un algorithme (sa ''</ins>[[<ins class=\"diffchange diffchange-inline\">Th\u00e9orie de </ins>la <ins class=\"diffchange diffchange-inline\">complexit\u00e9 (informatique th\u00e9orique)</ins>|<ins class=\"diffchange diffchange-inline\">complexit\u00e9 algorithmique</ins>]]''<ins class=\"diffchange diffchange-inline\">) consiste en la recherche de deux quantit\u00e9s importantes. La premi\u00e8re quantit\u00e9 </ins>est <ins class=\"diffchange diffchange-inline\">l\u2019\u00e9volution du nombre d\u2019instructions </ins>de <ins class=\"diffchange diffchange-inline\">base </ins>en <ins class=\"diffchange diffchange-inline\">fonction de la quantit\u00e9 de donn\u00e9es </ins>\u00e0 <ins class=\"diffchange diffchange-inline\">traiter (</ins>par exemple<ins class=\"diffchange diffchange-inline\">, pour </ins>un [[<ins class=\"diffchange diffchange-inline\">algorithme de tri</ins>]]<ins class=\"diffchange diffchange-inline\">, il s</ins>'<ins class=\"diffchange diffchange-inline\">agit du nombre de donn\u00e9es \u00e0 trier), que l\u2019on privil\u00e9giera sur le temps </ins>d'<ins class=\"diffchange diffchange-inline\">ex\u00e9cution mesur\u00e9 en secondes (car ce dernier d\u00e9pend de la machine sur laquelle </ins>l'<ins class=\"diffchange diffchange-inline\">algorithme s</ins>'<ins class=\"diffchange diffchange-inline\">ex\u00e9cute). La seconde quantit\u00e9 estim\u00e9e est la quantit\u00e9 de m\u00e9moire n\u00e9cessaire pour effectuer les calculs. Baser le calcul de la complexit\u00e9 d\u2019un algorithme sur le temps ou la quantit\u00e9 effective de m\u00e9moire qu\u2019un ordinateur particulier prend pour effectuer ledit algorithme ne permet pas de prendre en compte la </ins>structure <ins class=\"diffchange diffchange-inline\">interne de l\u2019algorithme, ni la particularit\u00e9 de l\u2019ordinateur : selon sa charge de travail, la vitesse de son processeur, la vitesse d\u2019acc\u00e8s aux donn\u00e9es, l\u2019ex\u00e9cution de l\u2019algorithme (</ins>qui <ins class=\"diffchange diffchange-inline\">peut faire intervenir le hasard) ou son organisation de la m\u00e9moire</ins>, <ins class=\"diffchange diffchange-inline\">le temps d\u2019ex\u00e9cution et </ins>la <ins class=\"diffchange diffchange-inline\">quantit\u00e9 de m\u00e9moire ne seront pas les m\u00eames</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">{{Article d\u00e9taill\u00e9|Th\u00e9orie des anneaux}}</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>[[<del class=\"diffchange diffchange-inline\">Fichier:Noether.jpg|thumb|left|[[Emmy Noether]] utilise </del>la <del class=\"diffchange diffchange-inline\">notion d'espace vectoriel pour \u00e9tudier les [[Anneau noeth\u00e9rien</del>|<del class=\"diffchange diffchange-inline\">anneaux]</del>] <del class=\"diffchange diffchange-inline\">portant maintenant son nom.</del>]<del class=\"diffchange diffchange-inline\">]</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Un exemple c\u00e9l\u00e8bre d</del>'<del class=\"diffchange diffchange-inline\">anneau disposant aussi d</del>'<del class=\"diffchange diffchange-inline\">une structure d'espace vectoriel </del>est <del class=\"diffchange diffchange-inline\">celui des [[polyn\u00f4me formel|polyn\u00f4mes]] \u00e0 coefficients dans un corps. Cet espace vectoriel, </del>de <del class=\"diffchange diffchange-inline\">dimension infinie, est largement utilis\u00e9 </del>en <del class=\"diffchange diffchange-inline\">alg\u00e8bre lin\u00e9aire, </del>\u00e0 <del class=\"diffchange diffchange-inline\">travers </del>par exemple <del class=\"diffchange diffchange-inline\">le [[polyn\u00f4me minimal d'</del>un <del class=\"diffchange diffchange-inline\">endomorphisme|polyn\u00f4me minimal]] ou </del>[[<del class=\"diffchange diffchange-inline\">polyn\u00f4me caract\u00e9ristique|caract\u00e9ristique</del>]]<del class=\"diffchange diffchange-inline\">. Le [[polyn\u00f4me d</del>'<del class=\"diffchange diffchange-inline\">endomorphisme|morphisme canonique]] entre les polyn\u00f4mes et les applications lin\u00e9aires </del>d'<del class=\"diffchange diffchange-inline\">un espace vectoriel est \u00e0 </del>l'<del class=\"diffchange diffchange-inline\">origine d</del>'<del class=\"diffchange diffchange-inline\">une </del>structure <del class=\"diffchange diffchange-inline\">d'alg\u00e8bre </del>qui <del class=\"diffchange diffchange-inline\">est un anneau</del>, <del class=\"diffchange diffchange-inline\">si </del>la <del class=\"diffchange diffchange-inline\">multiplication externe est ''oubli\u00e9e''</del>.</div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Cette m\u00e9thode permet </del>d'<del class=\"diffchange diffchange-inline\">\u00e9lucider la structure </del>de <del class=\"diffchange diffchange-inline\">certains anneaux</del>. <del class=\"diffchange diffchange-inline\">Tout anneau est </del>un <del class=\"diffchange diffchange-inline\">espace vectoriel sur ceux </del>de <del class=\"diffchange diffchange-inline\">ses sous-anneaux qui sont </del>des <del class=\"diffchange diffchange-inline\">corps. L</del>'<del class=\"diffchange diffchange-inline\">espace vectoriel ressemble \u00e0 </del>la <del class=\"diffchange diffchange-inline\">structure d\u00e9velopp\u00e9e par Grassman. Cette remarque est utilis\u00e9e au d\u00e9but du {{s-|XX}}, </del>en <del class=\"diffchange diffchange-inline\">particulier par </del>[[<del class=\"diffchange diffchange-inline\">Emil Artin</del>]] et [[<del class=\"diffchange diffchange-inline\">Emmy Noether</del>]], <del class=\"diffchange diffchange-inline\">pour \u00e9lucider cette structure dans le cas des anneaux artiniens et </del>[[<del class=\"diffchange diffchange-inline\">Anneau noeth\u00e9rien</del>|<del class=\"diffchange diffchange-inline\">noeth\u00e9riens</del>]]<del class=\"diffchange diffchange-inline\">, qui sont </del>des <del class=\"diffchange diffchange-inline\">copies </del>de sous<del class=\"diffchange diffchange-inline\">-alg\u00e8bres sur un espace vectoriel construit sur </del>[[<del class=\"diffchange diffchange-inline\">sous-anneau</del>]] <del class=\"diffchange diffchange-inline\">qui s'av\u00e8re \u00eatre un corps</del>.</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Souvent, on examine les performances \u00ab au pire \u00bb, c'est-\u00e0-dire dans les configurations telles que le [[complexit\u00e9 en temps|temps </ins>d'<ins class=\"diffchange diffchange-inline\">ex\u00e9cution]] ou l'[[complexit\u00e9 en espace|espace m\u00e9moire]] est le plus grand. Il existe \u00e9galement un autre aspect de l'\u00e9valuation </ins>de <ins class=\"diffchange diffchange-inline\">l'efficacit\u00e9 d'un algorithme : les performances \u00ab en moyenne \u00bb</ins>. <ins class=\"diffchange diffchange-inline\">Cela suppose d'avoir </ins>un <ins class=\"diffchange diffchange-inline\">mod\u00e8le </ins>de <ins class=\"diffchange diffchange-inline\">la r\u00e9partition statistique </ins>des <ins class=\"diffchange diffchange-inline\">donn\u00e9es de l</ins>'<ins class=\"diffchange diffchange-inline\">algorithme, tandis que </ins>la <ins class=\"diffchange diffchange-inline\">mise </ins>en <ins class=\"diffchange diffchange-inline\">\u0153uvre des techniques d'analyse implique des m\u00e9thodes assez fines de </ins>[[<ins class=\"diffchange diffchange-inline\">analyse combinatoire|combinatoire</ins>]] et <ins class=\"diffchange diffchange-inline\">d'</ins>[[<ins class=\"diffchange diffchange-inline\">D\u00e9veloppement asymptotique|\u00e9valuation asymptotique</ins>]], <ins class=\"diffchange diffchange-inline\">utilisant en particulier les </ins>[[<ins class=\"diffchange diffchange-inline\">s\u00e9rie g\u00e9n\u00e9ratrice</ins>|<ins class=\"diffchange diffchange-inline\">s\u00e9ries g\u00e9n\u00e9ratrices</ins>]] <ins class=\"diffchange diffchange-inline\">et </ins>des <ins class=\"diffchange diffchange-inline\">m\u00e9thodes avanc\u00e9es d'[[analyse complexe]]. L'ensemble </ins>de <ins class=\"diffchange diffchange-inline\">ces m\u00e9thodes est regroup\u00e9 </ins>sous <ins class=\"diffchange diffchange-inline\">le nom de </ins>[[<ins class=\"diffchange diffchange-inline\">combinatoire analytique</ins>]].</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Un exemple est </del>la <del class=\"diffchange diffchange-inline\">g\u00e9n\u00e9ralisation d'un th\u00e9or\u00e8me de </del>[[<del class=\"diffchange diffchange-inline\">Joseph Wedderburn|Wedderburn</del>]] <del class=\"diffchange diffchange-inline\">par Artin et portant maintenant le nom </del>de <del class=\"diffchange diffchange-inline\">[[th\u00e9or\u00e8me d'Artin</del>-<del class=\"diffchange diffchange-inline\">Wedderburn]]. Il est important </del>en <del class=\"diffchange diffchange-inline\">[[Th\u00e9orie des anneaux|alg\u00e8bre non commutative]]</del>.</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">On trouvera dans l\u2019article sur </ins>la [[<ins class=\"diffchange diffchange-inline\">th\u00e9orie de la complexit\u00e9 des algorithmes</ins>]] <ins class=\"diffchange diffchange-inline\">d\u2019autres \u00e9valuations </ins>de <ins class=\"diffchange diffchange-inline\">la complexit\u00e9 qui vont en g\u00e9n\u00e9ral au-del\u00e0 des valeurs propos\u00e9es ci</ins>-<ins class=\"diffchange diffchange-inline\">dessus et qui classifient les probl\u00e8mes algorithmiques (plut\u00f4t que les algorithmes) </ins>en <ins class=\"diffchange diffchange-inline\">classes de complexit\u00e9</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">[[Anneau oppos\u00e9|Un lemme \u00e9l\u00e9mentaire]] permet par ailleurs d'interpr\u00e9ter le corps </del>des <del class=\"diffchange diffchange-inline\">[[quaternion]]s comme l</del>'<del class=\"diffchange diffchange-inline\">alg\u00e8bre des [[Repr\u00e9sentation </del>de <del class=\"diffchange diffchange-inline\">groupe#D\u00e9finitions|endomorphismes]] d</del>'<del class=\"diffchange diffchange-inline\">[[Quaternion#Repr\u00e9sentation des quaternions comme matrices 4x4 </del>de <del class=\"diffchange diffchange-inline\">nombres r\u00e9els</del>|<del class=\"diffchange diffchange-inline\">une repr\u00e9sentation r\u00e9elle </del>de <del class=\"diffchange diffchange-inline\">degr\u00e9 4]] du </del>[[<del class=\"diffchange diffchange-inline\">groupe des quaternions|groupe associ\u00e9</del>]].</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==== Quelques indications sur l\u2019efficacit\u00e9 </ins>des <ins class=\"diffchange diffchange-inline\">algorithmes et ses biais ====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">L</ins>'<ins class=\"diffchange diffchange-inline\">efficacit\u00e9 algorithmique n\u2019est souvent connue que de mani\u00e8re asymptotique, c\u2019est-\u00e0-dire pour </ins>de <ins class=\"diffchange diffchange-inline\">grandes valeurs du param\u00e8tre ''n'</ins>'<ins class=\"diffchange diffchange-inline\">. Lorsque ce param\u00e8tre est suffisamment petit, un algorithme de complexit\u00e9 asymptotique plus grande peut en pratique \u00eatre plus efficace. Ainsi, pour trier un tableau </ins>de <ins class=\"diffchange diffchange-inline\">{{nombre</ins>|<ins class=\"diffchange diffchange-inline\">30|lignes}} (c\u2019est un param\u00e8tre </ins>de <ins class=\"diffchange diffchange-inline\">petite taille), il est inutile d\u2019utiliser un algorithme \u00e9volu\u00e9 comme le </ins>[[<ins class=\"diffchange diffchange-inline\">tri rapide</ins>]] <ins class=\"diffchange diffchange-inline\">(l\u2019un des algorithmes de tri asymptotiquement les plus efficaces en moyenne) : l\u2019algorithme de tri le plus simple \u00e0 \u00e9crire sera suffisamment efficace</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">=== Th\u00e9orie </del>de <del class=\"diffchange diffchange-inline\">Galois ===</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Entre deux algorithmes informatiques </ins>de <ins class=\"diffchange diffchange-inline\">complexit\u00e9 identique, on utilisera celui dont l\u2019occupation m\u00e9moire est moindre</ins>. <ins class=\"diffchange diffchange-inline\">L\u2019analyse </ins>de <ins class=\"diffchange diffchange-inline\">la complexit\u00e9 algorithmique peut \u00e9galement servir </ins>\u00e0 <ins class=\"diffchange diffchange-inline\">\u00e9valuer l\u2019occupation m\u00e9moire d\u2019un algorithme</ins>. <ins class=\"diffchange diffchange-inline\">Enfin, le choix d\u2019un algorithme plut\u00f4t qu\u2019un autre doit se faire en fonction des donn\u00e9es que l\u2019on s\u2019attend </ins>\u00e0 <ins class=\"diffchange diffchange-inline\">lui fournir </ins>en <ins class=\"diffchange diffchange-inline\">entr\u00e9e</ins>. <ins class=\"diffchange diffchange-inline\">Ainsi, le [[tri rapide</ins>]]<ins class=\"diffchange diffchange-inline\">, lorsque l\u2019on choisit le premier \u00e9l\u00e9ment comme pivot, se comporte </ins>de <ins class=\"diffchange diffchange-inline\">fa\u00e7on d\u00e9sastreuse si on l\u2019applique \u00e0 une liste </ins>de <ins class=\"diffchange diffchange-inline\">valeurs d\u00e9j\u00e0 tri\u00e9e</ins>. <ins class=\"diffchange diffchange-inline\">Il n\u2019est donc pas judicieux </ins>de <ins class=\"diffchange diffchange-inline\">l\u2019utiliser si on pr\u00e9voit que le programme recevra en entr\u00e9e des listes d\u00e9j\u00e0 presque tri\u00e9es ou alors il faudra choisir le pivot al\u00e9atoirement</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">{{Article d\u00e9taill\u00e9|Th\u00e9orie de Galois}}</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">[[Fichier:Pentagone construit</del>.<del class=\"diffchange diffchange-inline\">png|thumb|La th\u00e9orie </del>de <del class=\"diffchange diffchange-inline\">Galois permet de d\u00e9terminer quels polygones r\u00e9guliers sont constructibles </del>\u00e0 <del class=\"diffchange diffchange-inline\">la r\u00e8gle et au compas</del>. <del class=\"diffchange diffchange-inline\">[[Construction du pentagone r\u00e9gulier </del>\u00e0 <del class=\"diffchange diffchange-inline\">la r\u00e8gle et au compas|Le pentagone </del>en <del class=\"diffchange diffchange-inline\">fait partie]]</del>.]]</div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">La th\u00e9orie </del>de <del class=\"diffchange diffchange-inline\">Galois contient </del>de <del class=\"diffchange diffchange-inline\">nombreux exemples d'espaces vectoriels. Elle consiste \u00e0 \u00e9tudier un corps comme un espace vectoriel sur un sous-corps</del>. <del class=\"diffchange diffchange-inline\">Ainsi chaque sous-corps permet </del>de <del class=\"diffchange diffchange-inline\">consid\u00e9rer la structure initiale comme un espace vectoriel particulier</del>.</div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Un exemple </del>d'<del class=\"diffchange diffchange-inline\">application est celui des figures </del>[[<del class=\"diffchange diffchange-inline\">construction \u00e0 la r\u00e8gle et au compas</del>|<del class=\"diffchange diffchange-inline\">constructible </del>\u00e0 la <del class=\"diffchange diffchange-inline\">r\u00e8gle et au compas</del>]]. <del class=\"diffchange diffchange-inline\">Ces points forment </del>un <del class=\"diffchange diffchange-inline\">corps disposant d</del>'<del class=\"diffchange diffchange-inline\">une structure d</del>'<del class=\"diffchange diffchange-inline\">espace vectoriel sur </del>les <del class=\"diffchange diffchange-inline\">nombres rationnels</del>. <del class=\"diffchange diffchange-inline\">Il est </del>de <del class=\"diffchange diffchange-inline\">dimension infinie et, </del>pour <del class=\"diffchange diffchange-inline\">chaque point</del>, le plus <del class=\"diffchange diffchange-inline\">petit sous-corps </del>le <del class=\"diffchange diffchange-inline\">contenant est </del>de <del class=\"diffchange diffchange-inline\">dimension finie \u00e9gale </del>\u00e0 <del class=\"diffchange diffchange-inline\">une </del>[[<del class=\"diffchange diffchange-inline\">Puissance de deux</del>|<del class=\"diffchange diffchange-inline\">puissance de 2</del>]]. <del class=\"diffchange diffchange-inline\">Un tel sous-corps </del>est <del class=\"diffchange diffchange-inline\">appel\u00e9 une </del>[[<del class=\"diffchange diffchange-inline\">tour d'extensions quadratiques</del>]]<del class=\"diffchange diffchange-inline\">. Cette propri\u00e9t\u00e9 </del>de <del class=\"diffchange diffchange-inline\">ces espaces vectoriels permet </del>de <del class=\"diffchange diffchange-inline\">r\u00e9soudre d</del>'<del class=\"diffchange diffchange-inline\">antiques conjectures comme la </del>[[<del class=\"diffchange diffchange-inline\">duplication du cube</del>]], <del class=\"diffchange diffchange-inline\">la </del>[[<del class=\"diffchange diffchange-inline\">trisection de l'angle</del>]] <del class=\"diffchange diffchange-inline\">ou la construction </del>d'un <del class=\"diffchange diffchange-inline\">[[polygone r\u00e9gulier]]</del>.</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">D'autres param\u00e8tres \u00e0 prendre en compte sont notamment :</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* les [[Biais (statistique)|biais]] intrins\u00e8ques (accept\u00e9s ou involontaires) de nombreux algorithmes peuvent tromper les utilisateurs ou syst\u00e8mes </ins>d'[[<ins class=\"diffchange diffchange-inline\">intelligence artificielle]], de ''[[machine learning]]'', de diagnostic informatique, m\u00e9canique, [[diagnostic m\u00e9dical</ins>|<ins class=\"diffchange diffchange-inline\">m\u00e9dical]], de pr\u00e9vision, de pr\u00e9vention, de sondages ou d'[[aide </ins>\u00e0 la <ins class=\"diffchange diffchange-inline\">d\u00e9cision]] (notamment pour les [[r\u00e9seaux sociaux]], l'\u00e9ducation [ex : [[parcoursup]</ins>] ]<ins class=\"diffchange diffchange-inline\">, la m\u00e9decine, la justice, la police, l'arm\u00e9e, la politique, l'embauche\u2026) prenant mal en compte ou pas du tous ces biais&lt;ref name=hertel2019/&gt;</ins>. <ins class=\"diffchange diffchange-inline\">En 2019, des chercheurs de [[T\u00e9l\u00e9com ParisTech]] ont produit </ins>un <ins class=\"diffchange diffchange-inline\">rapport inventoriant les principaux biais connus, et quelques pistes de rem\u00e9diation&lt;ref name=hertel2019&gt;Hertel &amp; Delattre V (2019) </ins>''<ins class=\"diffchange diffchange-inline\">[https://www.sciencesetavenir.fr/high-tech/intelligence-artificielle/</ins>les<ins class=\"diffchange diffchange-inline\">-algorithmes-sont-partout-leurs-biais-nous-trompent_131820#xtor=EPR-1-[SEAActu17h]-20190302 Les algorithmes sont partout, leurs biais de conception nous trompent]'' ; le 02.03</ins>.<ins class=\"diffchange diffchange-inline\">2019&lt;/ref&gt;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* la [[M\u00e9moire virtuelle#Principe de localit\u00e9|localit\u00e9]] </ins>de <ins class=\"diffchange diffchange-inline\">l\u2019algorithme. Par exemple </ins>pour <ins class=\"diffchange diffchange-inline\">un syst\u00e8me \u00e0 [[m\u00e9moire virtuelle]] ayant peu de [[m\u00e9moire vive]] (par rapport au nombre de donn\u00e9es \u00e0 traiter)</ins>, le <ins class=\"diffchange diffchange-inline\">[[tri rapide]] sera normalement </ins>plus <ins class=\"diffchange diffchange-inline\">efficace que le [[tri par tas]] car </ins>le <ins class=\"diffchange diffchange-inline\">premier ne passe qu\u2019une seule fois sur chaque \u00e9l\u00e9ment </ins>de <ins class=\"diffchange diffchange-inline\">la m\u00e9moire tandis que le second acc\u00e8de </ins>\u00e0 <ins class=\"diffchange diffchange-inline\">la m\u00e9moire de mani\u00e8re discontinue (ce qui augmente le risque de {{lang|en|''</ins>[[<ins class=\"diffchange diffchange-inline\">M\u00e9moire virtuelle#Swapping</ins>|<ins class=\"diffchange diffchange-inline\">swapping</ins>]]<ins class=\"diffchange diffchange-inline\">''}})</ins>.</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* certains algorithmes (ceux dont l'analyse de complexit\u00e9 </ins>est <ins class=\"diffchange diffchange-inline\">dite </ins>[[<ins class=\"diffchange diffchange-inline\">analyse amortie|amortie</ins>]]<ins class=\"diffchange diffchange-inline\">), pour certaines ex\u00e9cutions </ins>de <ins class=\"diffchange diffchange-inline\">l\u2019algorithme (cas marginaux), pr\u00e9sentent une complexit\u00e9 qui sera tr\u00e8s sup\u00e9rieure au cas moyen, mais ceci sera compens\u00e9 par des ex\u00e9cutions rendues efficaces du m\u00eame algorithme dans une suite d'invocations </ins>de <ins class=\"diffchange diffchange-inline\">cet algorithme.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* l</ins>'[[<ins class=\"diffchange diffchange-inline\">Analyse lisse d'algorithme</ins>]], <ins class=\"diffchange diffchange-inline\">qui mesure les performances des algorithmes sur les pires cas, mais avec une l\u00e9g\u00e8re perturbation des instances. Elle explique pourquoi certains algorithmes analys\u00e9s comme inefficaces autrement, sont en fait efficaces en pratique. L'</ins>[[<ins class=\"diffchange diffchange-inline\">algorithme du simplexe</ins>]] <ins class=\"diffchange diffchange-inline\">est un exemple </ins>d'un <ins class=\"diffchange diffchange-inline\">algorithme qui se comporte bien pour l'analyse lisse</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>L'<del class=\"diffchange diffchange-inline\">exemple historique </del>de <del class=\"diffchange diffchange-inline\">la th\u00e9orie </del>est <del class=\"diffchange diffchange-inline\">celui de </del>la <del class=\"diffchange diffchange-inline\">r\u00e9solution </del>d'<del class=\"diffchange diffchange-inline\">une </del>[[<del class=\"diffchange diffchange-inline\">\u00e9quation polynomiale</del>]]. <del class=\"diffchange diffchange-inline\">Le </del>[[<del class=\"diffchange diffchange-inline\">Th\u00e9or\u00e8me </del>d'<del class=\"diffchange diffchange-inline\">Abel </del>(<del class=\"diffchange diffchange-inline\">alg\u00e8bre</del>)<del class=\"diffchange diffchange-inline\">|th\u00e9or\u00e8me d</del>'<del class=\"diffchange diffchange-inline\">Abel</del>]] <del class=\"diffchange diffchange-inline\">donne </del>une [[<del class=\"diffchange diffchange-inline\">\u00c9quivalence logique|condition n\u00e9cessaire et suffisante</del>]] de <del class=\"diffchange diffchange-inline\">r\u00e9solution </del>par [[<del class=\"diffchange diffchange-inline\">Racine </del>d'un <del class=\"diffchange diffchange-inline\">nombre#Racines d'</del>un <del class=\"diffchange diffchange-inline\">complexe|radicaux]]</del>. Les <del class=\"diffchange diffchange-inline\">espaces vectoriels utilis\u00e9s ont pour \u00e9l\u00e9ments ceux du plus petit corps </del>''<del class=\"diffchange diffchange-inline\">L</del>'' <del class=\"diffchange diffchange-inline\">contenant tous </del>les <del class=\"diffchange diffchange-inline\">coefficients du polyn\u00f4me ainsi </del>que <del class=\"diffchange diffchange-inline\">ses racines et le corps sous</del>-<del class=\"diffchange diffchange-inline\">jacent </del>est un <del class=\"diffchange diffchange-inline\">sous-corps </del>''<del class=\"diffchange diffchange-inline\">K'</del>' du <del class=\"diffchange diffchange-inline\">premier contenant tous les coefficients. Le </del>[[<del class=\"diffchange diffchange-inline\">groupe de Galois</del>]] est <del class=\"diffchange diffchange-inline\">compos\u00e9 des automorphismes du corps </del>''<del class=\"diffchange diffchange-inline\">L</del>'' <del class=\"diffchange diffchange-inline\">qui laissent invariant le corps </del>''<del class=\"diffchange diffchange-inline\">K'</del>'. <del class=\"diffchange diffchange-inline\">Ces automorphismes sont </del>en <del class=\"diffchange diffchange-inline\">nombre fini </del>et <del class=\"diffchange diffchange-inline\">sont </del>des [[<del class=\"diffchange diffchange-inline\">Application lin\u00e9aire#Cas particuliers|automorphismes du </del>''<del class=\"diffchange diffchange-inline\">K</del>''<del class=\"diffchange diffchange-inline\">-espace vectoriel</del>]] ''<del class=\"diffchange diffchange-inline\">L</del>''<del class=\"diffchange diffchange-inline\">. L</del>'<del class=\"diffchange diffchange-inline\">\u00e9l\u00e9ment cl\u00e9 </del>de [[<del class=\"diffchange diffchange-inline\">Th\u00e9or\u00e8me d</del>'<del class=\"diffchange diffchange-inline\">Abel </del>(<del class=\"diffchange diffchange-inline\">alg\u00e8bre</del>)<del class=\"diffchange diffchange-inline\">#D\u00e9monstration du th\u00e9or\u00e8me </del>de <del class=\"diffchange diffchange-inline\">Galois|la d\u00e9monstration</del>]] <del class=\"diffchange diffchange-inline\">montre que l'\u00e9quation est r\u00e9soluble seulement si ces automorphismes sont </del>[[<del class=\"diffchange diffchange-inline\">Diagonalisation</del>|<del class=\"diffchange diffchange-inline\">diagonalisables</del>]].</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Approches pratiques ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>L'<ins class=\"diffchange diffchange-inline\">algorithmique a d\u00e9velopp\u00e9 quelques strat\u00e9gies pour r\u00e9soudre les probl\u00e8mes :</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[algorithme glouton]] : un premier algorithme peut souvent \u00eatre propos\u00e9 en \u00e9tudiant le probl\u00e8me tr\u00e8s progressivement : on r\u00e9sout chaque sous-probl\u00e8me localement en esp\u00e9rant que l'ensemble </ins>de <ins class=\"diffchange diffchange-inline\">leurs r\u00e9sultats composera bien une solution du probl\u00e8me global. On parle alors d'algorithme glouton. L'algorithme glouton n'</ins>est <ins class=\"diffchange diffchange-inline\">souvent qu'une premi\u00e8re \u00e9tape dans </ins>la <ins class=\"diffchange diffchange-inline\">r\u00e9daction </ins>d'<ins class=\"diffchange diffchange-inline\">un algorithme plus performant ;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* </ins>[[<ins class=\"diffchange diffchange-inline\">Diviser pour r\u00e9gner (informatique)|diviser pour r\u00e9gner</ins>]] <ins class=\"diffchange diffchange-inline\">: pour am\u00e9liorer les performances des algorithmes, une technique usuelle consiste \u00e0 diviser les donn\u00e9es d'un probl\u00e8me en sous-ensembles de tailles plus petites, jusqu'\u00e0 obtenir des donn\u00e9es que l'algorithme pourra traiter au cas par cas</ins>. <ins class=\"diffchange diffchange-inline\">Une seconde \u00e9tape dans ces algorithmes consiste \u00e0 \u00ab fusionner \u00bb les r\u00e9sultats partiels pour obtenir une solution globale. Ces algorithmes sont souvent associ\u00e9s \u00e0 la r\u00e9cursivit\u00e9 ;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* </ins>[[<ins class=\"diffchange diffchange-inline\">recherche exhaustive]] (ou combinatoire) : une m\u00e9thode utilisant l'\u00e9norme puissance de calcul des ordinateurs consiste \u00e0 regarder tous les cas possibles. Cela n'est pour autant possible que dans certains cas particuliers (la combinatoire est souvent plus forte que l'\u00e9norme puissance des ordinateurs, aussi \u00e9norme soit-elle) ;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* d\u00e9composition ''top-down'' / ''bottom-up'' : (d\u00e9composition descendante, d\u00e9composition remontante) les d\u00e9compositions ''top-down'' consistent \u00e0 essayer de d\u00e9composer le probl\u00e8me en sous-probl\u00e8mes \u00e0 r\u00e9soudre successivement, la d\u00e9composition allant jusqu'\u00e0 des probl\u00e8mes triviaux faciles \u00e0 r\u00e9soudre. L'algorithme global est alors donn\u00e9 par la compos\u00e9e des algorithmes d\u00e9finis au cours de la d\u00e9composition. La d\u00e9marche ''bottom-up'' est la d\u00e9marche inverse, elle consiste \u00e0 partir </ins>d'<ins class=\"diffchange diffchange-inline\">algorithmes simples, ne r\u00e9solvant qu'une \u00e9tape du probl\u00e8me, pour essayer de les composer pour obtenir un algorithme global ;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* pr\u00e9-traitement / post-traitement : parfois, certains algorithmes comportent une ou deux phases identifi\u00e9es comme des pr\u00e9-traitements </ins>(<ins class=\"diffchange diffchange-inline\">\u00e0 faire avant l'algorithme principal), ou post-traitement (\u00e0 faire apr\u00e8s l'algorithme principal</ins>)<ins class=\"diffchange diffchange-inline\">, pour simplifier l'\u00e9criture de l</ins>'<ins class=\"diffchange diffchange-inline\">algorithme g\u00e9n\u00e9ral ;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[programmation dynamique</ins>]] <ins class=\"diffchange diffchange-inline\">: elle s'applique lorsque le probl\u00e8me d'optimisation est compos\u00e9 de plusieurs sous-probl\u00e8mes de m\u00eame nature, et qu'une solution optimale du probl\u00e8me global s'obtient \u00e0 partir de solutions optimales des sous-probl\u00e8mes.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== Les heuristiques ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{article d\u00e9taill\u00e9|Algorithme de Las Vegas|Algorithme de Monte-Carlo}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Pour certains probl\u00e8mes, les algorithmes ont une complexit\u00e9 beaucoup trop grande pour obtenir un r\u00e9sultat en temps raisonnable, m\u00eame si l\u2019on pouvait utiliser </ins>une <ins class=\"diffchange diffchange-inline\">puissance de calcul ph\u00e9nom\u00e9nale. On est donc amen\u00e9 \u00e0 rechercher la solution de fa\u00e7on non syst\u00e9matique (</ins>[[<ins class=\"diffchange diffchange-inline\">algorithme de Las Vegas</ins>]]<ins class=\"diffchange diffchange-inline\">) ou </ins>de <ins class=\"diffchange diffchange-inline\">se contenter d'une solution la plus proche possible d\u2019une solution optimale en proc\u00e9dant </ins>par <ins class=\"diffchange diffchange-inline\">essais successifs ([[algorithme de Monte-Carlo]]). Puisque toutes les combinaisons ne peuvent \u00eatre essay\u00e9es, certains choix strat\u00e9giques doivent \u00eatre faits. Ces choix, g\u00e9n\u00e9ralement tr\u00e8s d\u00e9pendants du probl\u00e8me trait\u00e9, constituent ce qu\u2019on appelle une </ins>[[<ins class=\"diffchange diffchange-inline\">heuristique (math\u00e9matiques)|heuristique]]. Le but d\u2019une heuristique n'est donc pas </ins>d'<ins class=\"diffchange diffchange-inline\">essayer toutes les combinaisons possibles, mais de trouver une solution en </ins>un <ins class=\"diffchange diffchange-inline\">temps raisonnable et par </ins>un <ins class=\"diffchange diffchange-inline\">autre moyen, par exemple en proc\u00e9dant \u00e0 des tirages al\u00e9atoires. La solution peut \u00eatre exacte (Las Vegas) ou approch\u00e9e (Monte-Carlo)</ins>. Les ''<ins class=\"diffchange diffchange-inline\">algorithmes d'Atlantic City</ins>'' <ins class=\"diffchange diffchange-inline\">quant \u00e0 eux donnent de fa\u00e7on probablement efficace une r\u00e9ponse probablement juste (disons avec une chance sur cent millions de se tromper) \u00e0 la question pos\u00e9e.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">C\u2019est ainsi que </ins>les <ins class=\"diffchange diffchange-inline\">programmes de [[\u00c9checs|jeu d\u2019\u00e9checs]] ou de [[jeu de go]] (pour ne citer </ins>que <ins class=\"diffchange diffchange-inline\">ceux</ins>-<ins class=\"diffchange diffchange-inline\">l\u00e0) font appel de mani\u00e8re tr\u00e8s fr\u00e9quente \u00e0 des heuristiques qui mod\u00e9lisent l\u2019exp\u00e9rience d\u2019un joueur. Certains [[Logiciel antivirus|logiciels antivirus]] se basent \u00e9galement sur des heuristiques pour reconna\u00eetre des [[virus informatique]]s non r\u00e9pertori\u00e9s dans leur base, en s\u2019appuyant sur des ressemblances avec des virus connus, c'</ins>est un <ins class=\"diffchange diffchange-inline\">exemple d</ins>'<ins class=\"diffchange diffchange-inline\">algorithme d</ins>'<ins class=\"diffchange diffchange-inline\">Atlantic City. De m\u00eame le [[probl\u00e8me SAT]] qui est l</ins>'<ins class=\"diffchange diffchange-inline\">arch\u00e9type </ins>du [[<ins class=\"diffchange diffchange-inline\">probl\u00e8me NP-complet</ins>]] <ins class=\"diffchange diffchange-inline\">donc tr\u00e8s difficile </ins>est <ins class=\"diffchange diffchange-inline\">r\u00e9solu de [[Probl\u00e8me SAT#Algorithmes de SAT|fa\u00e7on pratique et efficace par la mise au point d</ins>'<ins class=\"diffchange diffchange-inline\">heuristiques]]&lt;ref&gt;{{en}} [[Moshe Vardi]], </ins>''<ins class=\"diffchange diffchange-inline\">{{Langue|en|Boolean Satisfiability: Theory and Engineering}}</ins>'' <ins class=\"diffchange diffchange-inline\">[http://cacm.acm.org/magazines/2014/3/172516-boolean-satisfiability/fulltext (Communications of the ACM, Vol. 57 Nos. 3, p. 5)].&lt;/ref&gt;.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Exemples d\u2019algorithmes, de probl\u00e8mes, d</ins>'<ins class=\"diffchange diffchange-inline\">applications ou domaines d</ins>'<ins class=\"diffchange diffchange-inline\">application ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Il existe un certain nombre d\u2019algorithmes classiques, utilis\u00e9s pour r\u00e9soudre des probl\u00e8mes ou plus simplement pour illustrer des m\u00e9thodes de programmation</ins>. <ins class=\"diffchange diffchange-inline\">On se r\u00e9f\u00e9rera aux articles suivants pour de plus amples d\u00e9tails (voir aussi [[liste des algorithmes]]) :</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* algorithmes ou probl\u00e8mes classiques (du plus simple ou plus complexe) :</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** \u00e9change, ou comment \u00e9changer les valeurs de deux variables : probl\u00e8me classique illustrant la notion de variable informatique (voir aussi [[Structure de donn\u00e9es]]),</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** algorithmes de recherche, ou comment retrouver une information dans un ensemble structur\u00e9 ou non (par exemple [[Recherche dichotomique]]),</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** [[algorithme de tri]], ou comment trier un ensemble de nombres le plus rapidement possible ou </ins>en <ins class=\"diffchange diffchange-inline\">utilisant le moins de ressources possible,</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** [[probl\u00e8me du voyageur de commerce]], [[probl\u00e8me du sac \u00e0 dos]], [[probl\u00e8me SAT]] </ins>et <ins class=\"diffchange diffchange-inline\">autres algorithmes ou approximations de solutions pour les probl\u00e8mes combinatoires difficiles (dit NP-complets) ;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* algorithmes ou probl\u00e8mes illustrant la programmation r\u00e9cursive (voir aussi [[algorithme r\u00e9cursif]]) :</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">**[[tours de Hano\u00ef]],</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** [[huit dames]], placer huit dames sur un \u00e9chiquier sans qu\u2019elles puissent se prendre entre elles,</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** [[suite de Conway]],</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** algorithme de dessins r\u00e9cursifs ([[fractale]]) pour le [[Tapis de Sierpi\u0144ski]], la [[Courbe du dragon]], le [[Flocon de Koch]]\u2026 ;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* algorithmes dans le domaine </ins>des <ins class=\"diffchange diffchange-inline\">math\u00e9matiques :</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** calcul de la </ins>[[<ins class=\"diffchange diffchange-inline\">factorielle]] d</ins>'<ins class=\"diffchange diffchange-inline\">un nombre, de la [[Fonction d</ins>'<ins class=\"diffchange diffchange-inline\">Ackermann]] ou de la [[suite de Fibonacci]],</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** [[algorithme du simplexe]], qui minimise une fonction lin\u00e9aire de variables r\u00e9elles soumises \u00e0 des contraintes lin\u00e9aires,</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** [[fraction continue d</ins>'<ins class=\"diffchange diffchange-inline\">un nombre quadratique]], permettant d</ins>'<ins class=\"diffchange diffchange-inline\">extraire une [[racine carr\u00e9e]], cas particulier de la [[m\u00e9thode de Newton</ins>]]<ins class=\"diffchange diffchange-inline\">,</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** dans le domaine de l</ins>'<ins class=\"diffchange diffchange-inline\">alg\u00e8bre : l</ins>'<ins class=\"diffchange diffchange-inline\">[[unification|algorithme d</ins>'<ins class=\"diffchange diffchange-inline\">unification]], le calcul d</ins>'<ins class=\"diffchange diffchange-inline\">une [[bases de Gr\u00f6bner|base de Gr\u00f6bner]] d</ins>'<ins class=\"diffchange diffchange-inline\">un id\u00e9al de polyn\u00f4me et plus g\u00e9n\u00e9ralement presque toutes les m\u00e9thodes de [[calcul symbolique]],</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** en [[Th\u00e9orie des graphes#Aspect algorithmique|th\u00e9orie des graphes]] qui donne lieu \u00e0 </ins>de <ins class=\"diffchange diffchange-inline\">nombreux algorithmes,</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** </ins>[[<ins class=\"diffchange diffchange-inline\">test de primalit\u00e9]] ;</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* algorithmes pour et dans le domaine de l</ins>'<ins class=\"diffchange diffchange-inline\">informatique :</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** [[cryptologie]] et [[compression de donn\u00e9es]],</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** [[informatique musicale]],</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** [[algorithme g\u00e9n\u00e9tique]] en [[informatique d\u00e9cisionnelle]],</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** analyse et compilation des langages formels </ins>(<ins class=\"diffchange diffchange-inline\">voir [[Compilateur]] et [[Interpr\u00e8te (informatique)]]</ins>)<ins class=\"diffchange diffchange-inline\">,</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">** [[allocation </ins>de <ins class=\"diffchange diffchange-inline\">m\u00e9moire</ins>]] <ins class=\"diffchange diffchange-inline\">(</ins>[[<ins class=\"diffchange diffchange-inline\">ramasse-miettes (informatique)</ins>|<ins class=\"diffchange diffchange-inline\">ramasse-miettes</ins>]]<ins class=\"diffchange diffchange-inline\">)</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><div>== Notes et r\u00e9f\u00e9rences ==</div></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><div>== Notes et r\u00e9f\u00e9rences ==</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange\"></del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><div>{{R\u00e9f\u00e9rences}}</div></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><div>{{R\u00e9f\u00e9rences}}</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>== <del class=\"diffchange diffchange-inline\">Bibliographie ==</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>== <ins class=\"diffchange diffchange-inline\">Annexes </ins>==</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* Vincent Blanloeil, ''Une introduction moderne \u00e0 l\u2019alg\u00e8bre lin\u00e9aire'', Ellipses, 2012</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* Roger Mansuy et Rached Mneimn\u00e9, ''Alg\u00e8bre lin\u00e9aire - R\u00e9duction des endomorphismes'', Vuibert, 2012</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>\u00a0</div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">== Voir aussi </del>==</div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><div>{{Autres projets</div></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><div>{{Autres projets</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>|<del class=\"diffchange diffchange-inline\">commons</del>=<del class=\"diffchange diffchange-inline\">Category:Linear algebra</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>|<ins class=\"diffchange diffchange-inline\">wiktionary</ins>=<ins class=\"diffchange diffchange-inline\">algorithmie</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>|<del class=\"diffchange diffchange-inline\">wikt</del>=<del class=\"diffchange diffchange-inline\">alg\u00e8bre lin\u00e9aire</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>|<ins class=\"diffchange diffchange-inline\">wikiversity</ins>=<ins class=\"diffchange diffchange-inline\">Algorithmique</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>|<del class=\"diffchange diffchange-inline\">b</del>=<del class=\"diffchange diffchange-inline\">Alg\u00e8bre lin\u00e9aire</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>|<ins class=\"diffchange diffchange-inline\">wikibooks</ins>=<ins class=\"diffchange diffchange-inline\">Algorithmique imp\u00e9rative</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">|v=Alg\u00e8bre lin\u00e9aire</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><div>}}</div></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><div>}}</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange\"></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange\">=== Bibliographie ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange\">* {{Ouvrage|langue=en|auteur1=[[Donald Knuth|Donald E. Knuth]]|titre=[[The Art of Computer Programming]]|volume=2|titre volume=Seminumerical algorithms|lieu=Reading, Mass|\u00e9diteur=Addison-Wesley Pub. Co|ann\u00e9e=1973|pages totales=764|isbn=978-0-201-89684-8|isbn2=978-0-321-75104-1|oclc=781024586}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange\">* {{Algorithmique (Quercia)}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange\">* {{Cormen3fr}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange\">* {{Ouvrage|langue=fr|auteur1=Patrick Bosc|auteur2=Marc Guyomard|auteur3=Laurent Miclet|titre=Conception d'algorithmes|sous-titre=principes et 150 exercices corrig\u00e9s|lieu=Paris|\u00e9diteur=Eyrolles|ann\u00e9e=2019|pages totales=832|isbn=978-2-212-67728-7|bnf=456636375}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange\"></ins></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><div>=== Articles connexes ===</div></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><div>=== Articles connexes ===</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* [[<del class=\"diffchange diffchange-inline\">Propri\u00e9t\u00e9s m\u00e9triques des droites et plans</del>]]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>* [[<ins class=\"diffchange diffchange-inline\">Algorithme r\u00e9cursif]]</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>*[[<del class=\"diffchange diffchange-inline\">Alg\u00e8bre g\u00e9n\u00e9rale</del>]]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Algorithme r\u00e9parti]]</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* [[<del class=\"diffchange diffchange-inline\">Alg\u00e8bre multilin\u00e9aire</del>]]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Algorithme \u00e9mergent]]</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* [[<del class=\"diffchange diffchange-inline\">Loi </del>d'<del class=\"diffchange diffchange-inline\">inertie de Sylvester</del>]]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Algorithme adaptatif</ins>]]</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* [[<del class=\"diffchange diffchange-inline\">Optimisation lin\u00e9aire</del>]]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>* [[<ins class=\"diffchange diffchange-inline\">Algorithme d'approximation</ins>]]</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* [[<del class=\"diffchange diffchange-inline\">Algorithme de Bartels-Stewart</del>]]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>* [[<ins class=\"diffchange diffchange-inline\">Art algorithmique</ins>]]</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>* [[<ins class=\"diffchange diffchange-inline\">Liste </ins>d'<ins class=\"diffchange diffchange-inline\">algorithmes]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[M\u00e9taheuristique</ins>]]</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>* [[<ins class=\"diffchange diffchange-inline\">Recherche op\u00e9rationnelle</ins>]]</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>* [[<ins class=\"diffchange diffchange-inline\">Paradigme (programmation)</ins>]]</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><div>=== Liens externes ===</div></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><div>=== Liens externes ===</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* </del>{{<del class=\"diffchange diffchange-inline\">en}} [http://www.egwald.com/linearalgebra/index.php Linear Algebra] par Elmer G. Wiens</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>{{<ins class=\"diffchange diffchange-inline\">Liens</ins>}}</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* [https://web.archive.org/web/20101108172830/http://roso.epfl.ch/teaching.html Les cours du ROSO, dont de l'Alg\u00e8bre lin\u00e9aire]</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* [http://braise.univ-rennes1.fr/\u00a0 Braise : la base raisonn\u00e9e d'exercices de math\u00e9matiques et son chapitre sur l'Alg\u00e8bre lin\u00e9aire]</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* {{lien web</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\"> | url = https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\"> | titre = Essence of linear algebra</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\"> | auteur = 3Blue1Brown</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\"> | site = YouTube</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\"> | consult\u00e9 le = 2018-06-12</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\"> | langue = en</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">}} {{commentaire biblio|cha\u00eene dont le but est\u00a0 \u00ab d'animer les intuitions g\u00e9om\u00e9triques soustendant de nombreux sujets enseign\u00e9s dans les cours habituels d'alg\u00e8bre lin\u00e9aire. \u00bb</del>}}</div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>{{Palette|<del class=\"diffchange diffchange-inline\">Alg\u00e8bre lin\u00e9aire</del>|Domaines <del class=\"diffchange diffchange-inline\">des math\u00e9matiques</del>}}</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>{{Palette|<ins class=\"diffchange diffchange-inline\">Informatique th\u00e9orique</ins>|Domaines <ins class=\"diffchange diffchange-inline\">de l'informatique</ins>}}</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>{{Portail|<del class=\"diffchange diffchange-inline\">Alg\u00e8bre</del>}}</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>{{Portail|<ins class=\"diffchange diffchange-inline\">informatique th\u00e9orique</ins>}}</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">{{DEFAULTSORT</del>:<del class=\"diffchange diffchange-inline\">Algebre lineaire}}</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Cat\u00e9gorie:Algorithmique| ]]</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>[[Cat\u00e9gorie:<del class=\"diffchange diffchange-inline\">Alg\u00e8bre lin\u00e9aire| </del>]]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Cat\u00e9gorie</ins>:<ins class=\"diffchange diffchange-inline\">Nom d\u00e9riv\u00e9 d'un anthroponyme]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>[[Cat\u00e9gorie:<ins class=\"diffchange diffchange-inline\">Branche des math\u00e9matiques</ins>]]</div></td></tr>\n"
    }
}