{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Annonce\n",
    "La branche étudiante de l'IEEE propose, ce jeudi 25 février de 13 h 30 à 17 h 30, une formation à Mathematica pour donner les bases de ce logiciel. L'inscription est nécessaire.\n",
    "http://ieee.aees.be/fr/accueil/25-francais/activites/conferences/151-formation-mathematica"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Initialisation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pour que la division soit comme en Python 3:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from __future__ import division"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Importer toutes les fonctions et quelques variables de Sympy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "from sympy.abc import a,b,c,k,n,t,u,v,w,x,y,z\n",
    "init_printing(pretty_print=True, use_latex='mathjax')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Plusieurs choses ne sont pas dans les notes de cours"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculer le pgcd de $$p(x)=x^5 - 20x^4 + 140x^3 - 430x^2 + 579x - 270$$ et $$q(x)=x^6 - 25x^5 + 243x^4 - 1163x^3 + 2852x^2 - 3348x + 1440$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "p = x**5 - 20*x**4 + 140*x**3 - 430*x**2 + 579*x - 270\n",
    "q = x**6 - 25*x**5 + 243*x**4 - 1163*x**3 + 2852*x**2 - 3348*x + 1440"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$2$$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gcd(4,6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$x^{4} - 11 x^{3} + 41 x^{2} - 61 x + 30$$"
      ],
      "text/plain": [
       " 4       3       2            \n",
       "x  - 11⋅x  + 41⋅x  - 61⋅x + 30"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gcd(p,q)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 7 Calcul différentiel et intégral"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 7.1 Limites"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from sympy.abc import x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "limit(1/x, x, 0, dir='+')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "oo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "limit(1/x, x, oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 7.2 Sommes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculer la somme des nombres 134,245,325,412,57."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$1173$$"
      ],
      "text/plain": [
       "1173"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum([134, 245, 325, 412, 57])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$x + 1173$$"
      ],
      "text/plain": [
       "x + 1173"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum([134, 245, 325, 412, 57, x])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "unsupported operand type(s) for +: 'Add' and 'list'",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-11-86a5c62c0901>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0msum\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m134\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m245\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m325\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m412\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m57\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m: unsupported operand type(s) for +: 'Add' and 'list'"
     ]
    }
   ],
   "source": [
    "sum([134, 245, 325, 412, 57, x, []])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Caluler la somme \n",
    "$$\\sum_{i=0}^n i$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{n^{2}}{2} + \\frac{n}{2}$$"
      ],
      "text/plain": [
       " 2    \n",
       "n    n\n",
       "── + ─\n",
       "2    2"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy.abc import i\n",
    "summation(i, (i,0,n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$2733212496$$"
      ],
      "text/plain": [
       "2733212496"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "summation(i**2, (i,0,2016))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculer la somme \n",
    "$$\\sum_{k=1}^\\infty {1 \\over k^6}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{\\pi^{6}}{945}$$"
      ],
      "text/plain": [
       "  6\n",
       " π \n",
       "───\n",
       "945"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "summation(1/k**6, (k, 1, oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 7.3 Produit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculer le produit \n",
    "$$\\prod_{n=1}^{2016} 2n+1$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$88681693265196907499163469239476550564853122378387396482417722319651626280932694998619107844525910610144309559122551894980732139408143312661761883645919264260705088711652544578595655862325247056991856090477885623811455687509107532289485678013515618683233663234520742050426499182224022059404146624378236148389579895304901512047465252468694956402886456504619441770462935648427546637951550204459125650515834991758369772154977255476516184020950887749669713692614690936269584864240606840830985250977701249417769096710232207897793622073084657212018813435612412956021375699902577116582751559869740672841620939443332112690197509808415785107588314400061437555857670240671419122969999221610068010489837510035859777229099633682924406084346169979312716350735737810292353366433826143415421646441413441766974106142183063059486538005919232303045744981992247425420232796264389894503691419112631503379767104094141885360784536652380821290247902861934905104035941549799604212916928819253756904372416614557105032131542310812381391873876911500658899508503643697134861308971728520403272997131681966501367016379150836110832356261327229411023864969052882275330041272503996998966976124816903474837561701916192304641855289737520589817496693593155844780448609429064200713249961483538347140790801204770427863145753575578414953034174390768139697021818423128133972513775343022690212594014258039563506387819689813201125949476912932761373152902357373493922312079297970102799958579670496436146807738669815518102734473625368184944206619337734194365826369326622971929443455461276364911673241836359757170182938260937599393183746595377560002184190394042391925181427044148996461622941729074585115670170052870600203471111805146556911274460524576253559350975823890332839202962915418850676141857133603860718245758708763169974922080014517811155692514434432019921115301466484929935954426900824290906990543083986252202657978726413950190867922748613715796908513759080771217823419591639030214931483642065524265460586718893072566039257613074928317795418595446977235576448296038868226698972259693433594452453149146868138145635945214195146173612792446507176328111936773667960428580508497326794816615650059313994051289031219229205101857681237742310911737132688576344976115234595931047143948845439866564361132563384177724989366305075891293524600911759051587063849860151840251347235114396646188674183643097713207856781344741929472930834394685398715139635964758577692215074780125318615347368120171532215349999420952807161212215503158348441182431910920246304546742684534168056097698735347382332084935629564435875419236430214563469137907689004392308913669544749111233785676287674460096863476835859468181623114790147226682733127801993472174646552592834717086693083457830269346231863934130304721199127402050383531840116983779499487066051712572019895925880395204026038854975601730200649576392152915961639491696862949735190524391043943017208720714884301124291068719089510707058297462477922475426884398795316397109314897166903721377367685923048137964812380247123222261650485629894246475102155883165742023946503657419751118412116256288634439427517834589279875489876904849537606046682216931734429500974657518660178745235164139793298510033243048920271350080442254513685895567854037093198358166004894031451235521612808993253084023616343230297238196441948490803168492614635968008292778815562902732573044079981526615172538013694663116584698032872591454966860220625496763964881591031246822031452304104960066473708114170512183467911213552198208424724025990118990854869547693102343003173420307598793128726446441093916381429844826642489240327299478272874022882755882662211409086044779651875643655426023942978770549353909617376298499531491825707033841988426701961542348387544813353275891573697668119186063727124663157744080824741371146870772329870773921106938017045446052051570347743145842959013508827660887576794677162477690515746458209825769522919794117715301268666461680046843513547266279825321693535834937983621991290871460689487637142134153535696369267571323305136098065428901869601307155362312078430803247831479443719149001726448754433443177424379669737628270889525114113822103348841809065993568937028218488141380301836094528332110795738901881450676647482710916566139814945220035531718088731178471667688038855362199025925183628986978428937365400757390507275564663165303180969174781444809946167551484903884071828212056156724638909295010513209301888633661699902830835115802661373478693070859048293606155030660837069844587594785784905734964631329835011160238356321415577242772156683287205698281102391972895858646443837170120279829284852928626398144006297398547847639031711659781819047014050604948122483623170182543922920930873844283274816603456536970080209915889028233770766649367267759962062133903973128139898455667437507035423652765489160107766273006749774504399937029485076970911273480414777334455759910088073179471595479241521625233132530663249928294823035474049860634200104918089887805403687800590391699493941235966017150782302485322484899292563626835006607500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       "388549756017302006495763921529159616394916968629497351905243910439430172087207\n",
       "148843011242910687190895107070582974624779224754268843987953163971093148971669\n",
       "037213773676859230481379648123802471232222616504856298942464751021558831657420\n",
       "239465036574197511184121162562886344394275178345892798754898769048495376060466\n",
       "822169317344295009746575186601787452351641397932985100332430489202713500804422\n",
       "545136858955678540370931983581660048940314512355216128089932530840236163432302\n",
       "972381964419484908031684926146359680082927788155629027325730440799815266151725\n",
       "380136946631165846980328725914549668602206254967639648815910312468220314523041\n",
       "049600664737081141705121834679112135521982084247240259901189908548695476931023\n",
       "430031734203075987931287264464410939163814298448266424892403272994782728740228\n",
       "827558826622114090860447796518756436554260239429787705493539096173762984995314\n",
       "918257070338419884267019615423483875448133532758915736976681191860637271246631\n",
       "577440808247413711468707723298707739211069380170454460520515703477431458429590\n",
       "135088276608875767946771624776905157464582098257695229197941177153012686664616\n",
       "800468435135472662798253216935358349379836219912908714606894876371421341535356\n",
       "963692675713233051360980654289018696013071553623120784308032478314794437191490\n",
       "017264487544334431774243796697376282708895251141138221033488418090659935689370\n",
       "282184881413803018360945283321107957389018814506766474827109165661398149452200\n",
       "355317180887311784716676880388553621990259251836289869784289373654007573905072\n",
       "755646631653031809691747814448099461675514849038840718282120561567246389092950\n",
       "105132093018886336616999028308351158026613734786930708590482936061550306608370\n",
       "698445875947857849057349646313298350111602383563214155772427721566832872056982\n",
       "811023919728958586464438371701202798292848529286263981440062973985478476390317\n",
       "116597818190470140506049481224836231701825439229209308738442832748166034565369\n",
       "700802099158890282337707666493672677599620621339039731281398984556674375070354\n",
       "236527654891601077662730067497745043999370294850769709112734804147773344557599\n",
       "100880731794715954792415216252331325306632499282948230354740498606342001049180\n",
       "898878054036878005903916994939412359660171507823024853224848992925636268350066\n",
       "075003075111313573731598728279937988751462742299441491367291041628486017830841\n",
       "308141037707891486743280046581693910849026164818764042004471099598482534024039\n",
       "668356196439518455901514892533110049493182851008224580171426832352332892483003\n",
       "261972870482499130683868223835779042502896390769514569596590873038160042436861\n",
       "269652473611026918104409572000994088501793695716252508189266330955745347293686\n",
       "762973993829299172822786341083279797585203528418610746927830103436031446839253\n",
       "806594773635352637931280509185123265935583732126062828265210055128135891807718\n",
       "992780363129682350725110112738402414005450690370099929885827944588057425295774\n",
       "842547540507636740507079111844180856803345801517704258107689788970794497350702\n",
       "075644326695744693642955013778911113920578902942177098443363334357770667912931\n",
       "541535140451938971243928993253871665209448942187164414849645215633209893234569\n",
       "826430970079015100317135118457499667128763649263601715323362407428068953007052\n",
       "971104765133341366410029966735396063447212585702567835548017883421093254223714\n",
       "327303573805863128461144659296990075299045224811505180020275891862894902176375\n",
       "408939710621551948060073252576306808335351971869192803343205944219395887121840\n",
       "754152242495334592548215504389524813795112258630316033612415739092218922966321\n",
       "688296828539707803982738631516174283615114468542969224115568457343345155046284\n",
       "111443137711665743237698329654951411114144343983411999943200498819351196289062\n",
       "5"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "product(2*n+1, (n,1,2016))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 7.4 Calcul différentiel"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Calculer la dérivée de $$x^5+bx$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left ( b, \\quad x\\right )$$"
      ],
      "text/plain": [
       "(b, x)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b,x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$x$$"
      ],
      "text/plain": [
       "x"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(x**5+b*x, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$b + 5 x^{4}$$"
      ],
      "text/plain": [
       "       4\n",
       "b + 5⋅x "
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(x**5+b*x, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Calculer la dérivée de $$\\arcsin(x)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{1}{\\sqrt{- x^{2} + 1}}$$"
      ],
      "text/plain": [
       "      1      \n",
       "─────────────\n",
       "   __________\n",
       "  ╱    2     \n",
       "╲╱  - x  + 1 "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(asin(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 7.5 Calcul intégral"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Calculer l'intégrale $$\\int\\log(x)\\, dx$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$x \\log{\\left (x \\right )} - x$$"
      ],
      "text/plain": [
       "x⋅log(x) - x"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(log(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Calculer l'intégrale $$\\int a^x\\, dx$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\begin{cases} x & \\text{for}\\: \\log{\\left (a \\right )} = 0 \\\\\\frac{a^{x}}{\\log{\\left (a \\right )}} & \\text{otherwise} \\end{cases}$$"
      ],
      "text/plain": [
       "⎧  x     for log(a) = 0\n",
       "⎪                      \n",
       "⎪   x                  \n",
       "⎨  a                   \n",
       "⎪──────    otherwise   \n",
       "⎪log(a)                \n",
       "⎩                      "
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(a**x, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Calculer l'intégrale $$\\int x^a\\, dx$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\begin{cases} \\log{\\left (x \\right )} & \\text{for}\\: a = -1 \\\\\\frac{x^{a + 1}}{a + 1} & \\text{otherwise} \\end{cases}$$"
      ],
      "text/plain": [
       "⎧log(x)  for a = -1\n",
       "⎪                  \n",
       "⎪ a + 1            \n",
       "⎨x                 \n",
       "⎪──────  otherwise \n",
       "⎪a + 1             \n",
       "⎩                  "
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(x**a, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$2$$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "log(100, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "log?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Calculer l'intégrale \n",
    "$$\\int \\sec^2(x)\\,dx$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{\\sin{\\left (x \\right )}}{\\cos{\\left (x \\right )}}$$"
      ],
      "text/plain": [
       "sin(x)\n",
       "──────\n",
       "cos(x)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(sec(x)**2, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{x^{3} y^{2}}{6}$$"
      ],
      "text/plain": [
       " 3  2\n",
       "x ⋅y \n",
       "─────\n",
       "  6  "
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(integrate(x**2*y, x), y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\int_0^5\\int_0^2 x^2y\\,dx\\,dy$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{100}{3}$$"
      ],
      "text/plain": [
       "100/3"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(x**2*y, (x,0,2), (y,0,5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 7.6 Sommes, produits, dérivées et intégrales non évaluées"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "A = Sum(1/k**6, (k,1,oo))\n",
    "B = Product(2*n+1, (n,1,21))\n",
    "C = Derivative(asin(x), x)\n",
    "D = Integral(log(x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\sum_{k=1}^{\\infty} \\frac{1}{k^{6}} = \\frac{\\pi^{6}}{945}$$"
      ],
      "text/plain": [
       "  ∞           \n",
       " ____         \n",
       " ╲            \n",
       "  ╲   1      6\n",
       "   ╲  ──    π \n",
       "   ╱   6 = ───\n",
       "  ╱   k    945\n",
       " ╱            \n",
       " ‾‾‾‾         \n",
       "k = 1         "
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(A, A.doit())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\prod_{n=1}^{21} \\left(2 n + 1\\right) = 563862029680583509947946875$$"
      ],
      "text/plain": [
       "  21                                       \n",
       "┬───┬                                      \n",
       "│   │ 2⋅n + 1 = 563862029680583509947946875\n",
       "│   │                                      \n",
       "n = 1                                      "
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(B, B.doit())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{d}{d x} \\operatorname{asin}{\\left (x \\right )} = \\frac{1}{\\sqrt{- x^{2} + 1}}$$"
      ],
      "text/plain": [
       "d                   1      \n",
       "──(asin(x)) = ─────────────\n",
       "dx               __________\n",
       "                ╱    2     \n",
       "              ╲╱  - x  + 1 "
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(C, C.doit())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\int \\log{\\left (x \\right )}\\, dx = x \\log{\\left (x \\right )} - x$$"
      ],
      "text/plain": [
       "⌠                         \n",
       "⎮ log(x) dx = x⋅log(x) - x\n",
       "⌡                         "
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(D, D.doit())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 7.7 Développement en séries"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculer la série de Taylor de $\\tan(x)$ en $x_0=0$ d'ordre 14."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$x + \\frac{x^{3}}{3} + \\frac{2 x^{5}}{15} + \\frac{17 x^{7}}{315} + \\frac{62 x^{9}}{2835} + \\frac{1382 x^{11}}{155925} + \\frac{21844 x^{13}}{6081075} + \\mathcal{O}\\left(x^{14}\\right)$$"
      ],
      "text/plain": [
       "     3      5       7       9         11          13         \n",
       "    x    2⋅x    17⋅x    62⋅x    1382⋅x     21844⋅x      ⎛ 14⎞\n",
       "x + ── + ──── + ───── + ───── + ──────── + ───────── + O⎝x  ⎠\n",
       "    3     15     315     2835    155925     6081075          "
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(tan(x), x, 0, 14)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$x - \\frac{x^{3}}{6} + \\frac{x^{5}}{120} - \\frac{x^{7}}{5040} + \\frac{x^{9}}{362880} + \\mathcal{O}\\left(x^{10}\\right)$$"
      ],
      "text/plain": [
       "     3     5     7       9           \n",
       "    x     x     x       x       ⎛ 10⎞\n",
       "x - ── + ─── - ──── + ────── + O⎝x  ⎠\n",
       "    6    120   5040   362880         "
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(sin(x), x, 0, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 7.8 Équations différentielles"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{d}{d x} e^{x} = e^{x}$$"
      ],
      "text/plain": [
       "d ⎛ x⎞    x\n",
       "──⎝ℯ ⎠ = ℯ \n",
       "dx         "
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import E\n",
    "A = Derivative(E**x, x)\n",
    "Eq(A, A.doit())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Trouver une fonction $f(x)$ telle que ${d\\over dx} f(x) = f(x)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "f = Function('f')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$f{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "f(x)"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "eq = Eq(Derivative(f(x),x), f(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$f{\\left (x \\right )} = C_{1} e^{x}$$"
      ],
      "text/plain": [
       "           x\n",
       "f(x) = C₁⋅ℯ "
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsolve(eq)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Trouver une fonction $f(x)$ telle que ${d^2\\over dx^2} f(x) = -f(x)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "eq2 = Eq(Derivative(f(x),x,x), -f(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$f{\\left (x \\right )} = C_{1} \\sin{\\left (x \\right )} + C_{2} \\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "f(x) = C₁⋅sin(x) + C₂⋅cos(x)"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsolve(eq2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{d^{5}}{d x^{5}}  f{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "  5      \n",
       " d       \n",
       "───(f(x))\n",
       "  5      \n",
       "dx       "
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Derivative(f(x),x,x,x,x,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{d^{5}}{d x^{5}}  f{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "  5      \n",
       " d       \n",
       "───(f(x))\n",
       "  5      \n",
       "dx       "
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Derivative(f(x),x,5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{d^{5}}{d x^{5}}  f{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "  5      \n",
       " d       \n",
       "───(f(x))\n",
       "  5      \n",
       "dx       "
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x).diff(x,5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Résoudre$$y''-4y'+5y=0$$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$y{\\left (x \\right )} = \\left(C_{1} \\sin{\\left (x \\right )} + C_{2} \\cos{\\left (x \\right )}\\right) e^{2 x}$$"
      ],
      "text/plain": [
       "                                2⋅x\n",
       "y(x) = (C₁⋅sin(x) + C₂⋅cos(x))⋅ℯ   "
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy.abc import x,y\n",
    "eq = Eq(y(x).diff(x,x)-4*y(x).diff(x)+5*y(x),0)\n",
    "dsolve(eq, y(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 8 Algèbre linéaire"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 8.1 Définir une matrice"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Définir la matrice\n",
    "$$M=\\begin{bmatrix}\n",
    "2& 9& 3\\\\ 4& 5& 10\\\\ 2& 0& 3\n",
    "\\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}2 & 9 & 3\\\\4 & 5 & 10\\\\2 & 0 & 3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡2  9  3 ⎤\n",
       "⎢        ⎥\n",
       "⎢4  5  10⎥\n",
       "⎢        ⎥\n",
       "⎣2  0  3 ⎦"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Matrix([[2, 9, 3], [4, 5, 10], [2, 0, 3]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "M = Matrix(3,3,[2, 9, 3, 4, 5, 10, 2, 0, 3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Définir la matrice\n",
    "$$N=\\begin{bmatrix}\n",
    "2& 9& 3\\\\ 4& 5& 10\\\\ -6& -1& -17\n",
    "\\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}2 & 9 & 3\\\\4 & 5 & 10\\\\-6 & -1 & -17\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡2   9    3 ⎤\n",
       "⎢           ⎥\n",
       "⎢4   5   10 ⎥\n",
       "⎢           ⎥\n",
       "⎣-6  -1  -17⎦"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = Matrix(3,3,[2,9,3,4,5,10,-6,-1,-17]); N"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Définir le vecteur\n",
    "$$v=\\begin{bmatrix}\n",
    "5\\\\ 2\\\\ 1\n",
    "\\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}5\\\\2\\\\1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡5⎤\n",
       "⎢ ⎥\n",
       "⎢2⎥\n",
       "⎢ ⎥\n",
       "⎣1⎦"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = Matrix([5,2,1]); v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 8.2 Opérations de base"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}2 & 9 & 3\\\\4 & 5 & 10\\\\2 & 0 & 3\\end{matrix}\\right], \\quad \\left[\\begin{matrix}2 & 9 & 3\\\\4 & 5 & 10\\\\-6 & -1 & -17\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "⎛⎡2  9  3 ⎤, ⎡2   9    3 ⎤⎞\n",
       "⎜⎢        ⎥  ⎢           ⎥⎟\n",
       "⎜⎢4  5  10⎥  ⎢4   5   10 ⎥⎟\n",
       "⎜⎢        ⎥  ⎢           ⎥⎟\n",
       "⎝⎣2  0  3 ⎦  ⎣-6  -1  -17⎦⎠"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M, N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}4 & 18 & 6\\\\8 & 10 & 20\\\\-4 & -1 & -14\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡4   18   6 ⎤\n",
       "⎢           ⎥\n",
       "⎢8   10  20 ⎥\n",
       "⎢           ⎥\n",
       "⎣-4  -1  -14⎦"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M + N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}6 & 27 & 9\\\\12 & 15 & 30\\\\6 & 0 & 9\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡6   27  9 ⎤\n",
       "⎢          ⎥\n",
       "⎢12  15  30⎥\n",
       "⎢          ⎥\n",
       "⎣6   0   9 ⎦"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M * 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}22 & 60 & 45\\\\-32 & 51 & -108\\\\-14 & 15 & -45\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡22   60   45 ⎤\n",
       "⎢             ⎥\n",
       "⎢-32  51  -108⎥\n",
       "⎢             ⎥\n",
       "⎣-14  15  -45 ⎦"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M * N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}31\\\\40\\\\13\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡31⎤\n",
       "⎢  ⎥\n",
       "⎢40⎥\n",
       "⎢  ⎥\n",
       "⎣13⎦"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M * v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{5}{24} & - \\frac{3}{8} & \\frac{25}{24}\\\\\\frac{1}{9} & 0 & - \\frac{1}{9}\\\\- \\frac{5}{36} & \\frac{1}{4} & - \\frac{13}{36}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡              25 ⎤\n",
       "⎢5/24   -3/8   ── ⎥\n",
       "⎢              24 ⎥\n",
       "⎢                 ⎥\n",
       "⎢ 1/9    0    -1/9⎥\n",
       "⎢                 ⎥\n",
       "⎢             -13 ⎥\n",
       "⎢-5/36  1/4   ────⎥\n",
       "⎣              36 ⎦"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M ** -1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "Matrix det == 0; not invertible.",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-82-086bca9c1aa4>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mN\u001b[0m \u001b[0;34m**\u001b[0m \u001b[0;34m-\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m/Users/slabbe/Applications/sage-git/local/lib/python2.7/site-packages/sympy/core/decorators.pyc\u001b[0m in \u001b[0;36mbinary_op_wrapper\u001b[0;34m(self, other)\u001b[0m\n\u001b[1;32m    116\u001b[0m                     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    117\u001b[0m                         \u001b[0;32mreturn\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 118\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mother\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    119\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mbinary_op_wrapper\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    120\u001b[0m     \u001b[0;32mreturn\u001b[0m \u001b[0mpriority_decorator\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/slabbe/Applications/sage-git/local/lib/python2.7/site-packages/sympy/matrices/dense.pyc\u001b[0m in \u001b[0;36m__pow__\u001b[0;34m(self, other)\u001b[0m\n\u001b[1;32m    561\u001b[0m     \u001b[0;34m@\u001b[0m\u001b[0mcall_highest_priority\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'__rpow__'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    562\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m__pow__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mother\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 563\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0msuper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mDenseMatrix\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__pow__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mother\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    564\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    565\u001b[0m     \u001b[0;34m@\u001b[0m\u001b[0mcall_highest_priority\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'__pow__'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/slabbe/Applications/sage-git/local/lib/python2.7/site-packages/sympy/matrices/matrices.pyc\u001b[0m in \u001b[0;36m__pow__\u001b[0;34m(self, num)\u001b[0m\n\u001b[1;32m    537\u001b[0m             \u001b[0mn\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnum\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    538\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mn\u001b[0m \u001b[0;34m<\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 539\u001b[0;31m                 \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0minv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mn\u001b[0m   \u001b[0;31m# A**-2 = (A**-1)**2\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    540\u001b[0m             \u001b[0ma\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0meye\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcols\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    541\u001b[0m             \u001b[0ms\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/slabbe/Applications/sage-git/local/lib/python2.7/site-packages/sympy/matrices/matrices.pyc\u001b[0m in \u001b[0;36minv\u001b[0;34m(self, method, **kwargs)\u001b[0m\n\u001b[1;32m    307\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mmethod\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    308\u001b[0m             \u001b[0mkwargs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m'method'\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mmethod\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 309\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_eval_inverse\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    310\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    311\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0minv_mod\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mm\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/slabbe/Applications/sage-git/local/lib/python2.7/site-packages/sympy/matrices/dense.pyc\u001b[0m in \u001b[0;36m_eval_inverse\u001b[0;34m(self, **kwargs)\u001b[0m\n\u001b[1;32m    304\u001b[0m         \u001b[0mM\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_mutable\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    305\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mmethod\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;34m\"GE\"\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 306\u001b[0;31m             \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mM\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0minverse_GE\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0miszerofunc\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0miszerofunc\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    307\u001b[0m         \u001b[0;32melif\u001b[0m \u001b[0mmethod\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;34m\"LU\"\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    308\u001b[0m             \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mM\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0minverse_LU\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0miszerofunc\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0miszerofunc\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/slabbe/Applications/sage-git/local/lib/python2.7/site-packages/sympy/matrices/matrices.pyc\u001b[0m in \u001b[0;36minverse_GE\u001b[0;34m(self, iszerofunc)\u001b[0m\n\u001b[1;32m   2629\u001b[0m         \u001b[0mred\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mbig\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrref\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0miszerofunc\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0miszerofunc\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msimplify\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2630\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0many\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0miszerofunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mred\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mj\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mj\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mj\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mred\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrows\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2631\u001b[0;31m             \u001b[0;32mraise\u001b[0m \u001b[0mValueError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"Matrix det == 0; not invertible.\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2632\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2633\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_new\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mred\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mbig\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrows\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mValueError\u001b[0m: Matrix det == 0; not invertible."
     ]
    }
   ],
   "source": [
    "N ** -1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}2 & 4 & 2\\\\9 & 5 & 0\\\\3 & 10 & 3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡2  4   2⎤\n",
       "⎢        ⎥\n",
       "⎢9  5   0⎥\n",
       "⎢        ⎥\n",
       "⎣3  10  3⎦"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.transpose()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 8.3 Accéder aux coefficients"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}2 & 9 & 3\\\\4 & 5 & 10\\\\2 & 0 & 3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡2  9  3 ⎤\n",
       "⎢        ⎥\n",
       "⎢4  5  10⎥\n",
       "⎢        ⎥\n",
       "⎣2  0  3 ⎦"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$5$$"
      ],
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M[1,1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M[2,1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}2 & 9 & 3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "[2  9  3]"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.row(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}9\\\\5\\\\0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡9⎤\n",
       "⎢ ⎥\n",
       "⎢5⎥\n",
       "⎢ ⎥\n",
       "⎣0⎦"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.col(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}2 & 9 & 3\\\\4 & 5 & 10\\\\2 & 0 & 3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡2  9  3 ⎤\n",
       "⎢        ⎥\n",
       "⎢4  5  10⎥\n",
       "⎢        ⎥\n",
       "⎣2  0  3 ⎦"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "M[0,0] = pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 8.4 Construction de matrices particulières"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0  0  0  0  0  0⎤\n",
       "⎢                ⎥\n",
       "⎢0  0  0  0  0  0⎥\n",
       "⎢                ⎥\n",
       "⎢0  0  0  0  0  0⎥\n",
       "⎢                ⎥\n",
       "⎣0  0  0  0  0  0⎦"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zeros(4,6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\\\1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\\\1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  1  1  1  1  1  1  1⎤\n",
       "⎢                      ⎥\n",
       "⎢1  1  1  1  1  1  1  1⎥\n",
       "⎢                      ⎥\n",
       "⎣1  1  1  1  1  1  1  1⎦"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ones(3,8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 0 & 0 & 0 & 0\\\\0 & 1 & 0 & 0 & 0\\\\0 & 0 & 1 & 0 & 0\\\\0 & 0 & 0 & 1 & 0\\\\0 & 0 & 0 & 0 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  0  0  0  0⎤\n",
       "⎢             ⎥\n",
       "⎢0  1  0  0  0⎥\n",
       "⎢             ⎥\n",
       "⎢0  0  1  0  0⎥\n",
       "⎢             ⎥\n",
       "⎢0  0  0  1  0⎥\n",
       "⎢             ⎥\n",
       "⎣0  0  0  0  1⎦"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eye(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}3 & 0 & 0\\\\0 & 4 & 0\\\\0 & 0 & 5\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡3  0  0⎤\n",
       "⎢       ⎥\n",
       "⎢0  4  0⎥\n",
       "⎢       ⎥\n",
       "⎣0  0  5⎦"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diag(3,4,5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}3 & 0 & 0 & 0 & 0 & 0\\\\0 & 4 & 0 & 0 & 0 & 0\\\\0 & 0 & 5 & 0 & 0 & 0\\\\0 & 0 & 0 & \\pi & 9 & 3\\\\0 & 0 & 0 & 4 & 5 & 10\\\\0 & 0 & 0 & 2 & 0 & 3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡3  0  0  0  0  0 ⎤\n",
       "⎢                 ⎥\n",
       "⎢0  4  0  0  0  0 ⎥\n",
       "⎢                 ⎥\n",
       "⎢0  0  5  0  0  0 ⎥\n",
       "⎢                 ⎥\n",
       "⎢0  0  0  π  9  3 ⎥\n",
       "⎢                 ⎥\n",
       "⎢0  0  0  4  5  10⎥\n",
       "⎢                 ⎥\n",
       "⎣0  0  0  2  0  3 ⎦"
      ]
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diag(3,4,5,M)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 8.5 Matrice échelonnée réduite"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculer la forme échelonnée réduite de $M$ et $N$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}2 & 9 & 3\\\\4 & 5 & 10\\\\2 & 0 & 3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡2  9  3 ⎤\n",
       "⎢        ⎥\n",
       "⎢4  5  10⎥\n",
       "⎢        ⎥\n",
       "⎣2  0  3 ⎦"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{matrix}\\right], \\quad \\left [ 0, \\quad 1, \\quad 2\\right ]\\right )$$"
      ],
      "text/plain": [
       "⎛⎡1  0  0⎤, [0, 1, 2]⎞\n",
       "⎜⎢       ⎥           ⎟\n",
       "⎜⎢0  1  0⎥           ⎟\n",
       "⎜⎢       ⎥           ⎟\n",
       "⎝⎣0  0  1⎦           ⎠"
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.rref()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}2 & 9 & 3\\\\4 & 5 & 10\\\\-6 & -1 & -17\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡2   9    3 ⎤\n",
       "⎢           ⎥\n",
       "⎢4   5   10 ⎥\n",
       "⎢           ⎥\n",
       "⎣-6  -1  -17⎦"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}1 & 0 & \\frac{75}{26}\\\\0 & 1 & - \\frac{4}{13}\\\\0 & 0 & 0\\end{matrix}\\right], \\quad \\left [ 0, \\quad 1\\right ]\\right )$$"
      ],
      "text/plain": [
       "⎛⎡       75  ⎤        ⎞\n",
       "⎜⎢1  0   ──  ⎥, [0, 1]⎟\n",
       "⎜⎢       26  ⎥        ⎟\n",
       "⎜⎢           ⎥        ⎟\n",
       "⎜⎢0  1  -4/13⎥        ⎟\n",
       "⎜⎢           ⎥        ⎟\n",
       "⎝⎣0  0    0  ⎦        ⎠"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N.rref()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 8.6 Noyau"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculer le noyau des matrices $M$ et $N$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left [ \\right ]$$"
      ],
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.nullspace()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left [ \\left[\\begin{matrix}- \\frac{75}{26}\\\\\\frac{4}{13}\\\\1\\end{matrix}\\right]\\right ]$$"
      ],
      "text/plain": [
       "⎡⎡-75 ⎤⎤\n",
       "⎢⎢────⎥⎥\n",
       "⎢⎢ 26 ⎥⎥\n",
       "⎢⎢    ⎥⎥\n",
       "⎢⎢4/13⎥⎥\n",
       "⎢⎢    ⎥⎥\n",
       "⎣⎣ 1  ⎦⎦"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N.nullspace()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 8.7 Déterminant"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculer le déterminant des matrices $M$ et $N$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$72$$"
      ],
      "text/plain": [
       "72"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N.det()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 8.8 Polynôme caractéristique"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculer le polynôme caractérisque de la matrice $M$ et de $N$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy.abc import x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\operatorname{PurePoly}{\\left( x^{3} - 10 x^{2} - 11 x - 72, x, domain=\\mathbb{Z} \\right)}$$"
      ],
      "text/plain": [
       "PurePoly(x**3 - 10*x**2 - 11*x - 72, x, domain='ZZ')"
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.charpoly(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$x^{3} - 10 x^{2} - 11 x - 72$$"
      ],
      "text/plain": [
       " 3       2            \n",
       "x  - 10⋅x  - 11⋅x - 72"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.charpoly(x).as_expr()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$x^{3} + 10 x^{2} - 117 x$$"
      ],
      "text/plain": [
       " 3       2        \n",
       "x  + 10⋅x  - 117⋅x"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N.charpoly(x).as_expr()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "### 8.9 Valeurs propres et vecteurs propres"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Calculer les valeurs propres et vecteurs propres de\n",
    "$$K=\\begin{bmatrix}\n",
    "93& 27& -57\\\\ -40& 180& -140\\\\ -15& 27& 51\n",
    "\\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}93 & 27 & -57\\\\-40 & 180 & -140\\\\-15 & 27 & 51\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡93   27   -57 ⎤\n",
       "⎢              ⎥\n",
       "⎢-40  180  -140⎥\n",
       "⎢              ⎥\n",
       "⎣-15  27    51 ⎦"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K = Matrix(3,3,[93,27,-57,-40,180,-140,-15,27,51])\n",
    "K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left \\{ 90 : 1, \\quad 108 : 1, \\quad 126 : 1\\right \\}$$"
      ],
      "text/plain": [
       "{90: 1, 108: 1, 126: 1}"
      ]
     },
     "execution_count": 122,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K.eigenvals()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left [ \\left ( 90, \\quad 1, \\quad \\left [ \\left[\\begin{matrix}1\\\\2\\\\1\\end{matrix}\\right]\\right ]\\right ), \\quad \\left ( 108, \\quad 1, \\quad \\left [ \\left[\\begin{matrix}\\frac{9}{5}\\\\1\\\\0\\end{matrix}\\right]\\right ]\\right ), \\quad \\left ( 126, \\quad 1, \\quad \\left [ \\left[\\begin{matrix}1\\\\\\frac{10}{3}\\\\1\\end{matrix}\\right]\\right ]\\right )\\right ]$$"
      ],
      "text/plain": [
       "⎡⎛90, 1, ⎡⎡1⎤⎤⎞, ⎛108, 1, ⎡⎡9/5⎤⎤⎞, ⎛126, 1, ⎡⎡ 1  ⎤⎤⎞⎤\n",
       "⎢⎜       ⎢⎢ ⎥⎥⎟  ⎜        ⎢⎢   ⎥⎥⎟  ⎜        ⎢⎢    ⎥⎥⎟⎥\n",
       "⎢⎜       ⎢⎢2⎥⎥⎟  ⎜        ⎢⎢ 1 ⎥⎥⎟  ⎜        ⎢⎢10/3⎥⎥⎟⎥\n",
       "⎢⎜       ⎢⎢ ⎥⎥⎟  ⎜        ⎢⎢   ⎥⎥⎟  ⎜        ⎢⎢    ⎥⎥⎟⎥\n",
       "⎣⎝       ⎣⎣1⎦⎦⎠  ⎝        ⎣⎣ 0 ⎦⎦⎠  ⎝        ⎣⎣ 1  ⎦⎦⎠⎦"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K.eigenvects()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En général, les racines peuvent être plus compliquées:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left \\{ \\frac{10}{3} + \\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{2 \\sqrt{11523}}{3} + \\frac{2467}{27}} + \\frac{133}{9 \\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{2 \\sqrt{11523}}{3} + \\frac{2467}{27}}} : 1, \\quad \\frac{10}{3} + \\frac{133}{9 \\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{2 \\sqrt{11523}}{3} + \\frac{2467}{27}}} + \\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{2 \\sqrt{11523}}{3} + \\frac{2467}{27}} : 1, \\quad \\frac{133}{9 \\sqrt[3]{\\frac{2 \\sqrt{11523}}{3} + \\frac{2467}{27}}} + \\frac{10}{3} + \\sqrt[3]{\\frac{2 \\sqrt{11523}}{3} + \\frac{2467}{27}} : 1\\right \\}$$"
      ],
      "text/plain": [
       "⎧                          ____________________                               \n",
       "⎪     ⎛        ___  ⎞     ╱     _______                                       \n",
       "⎪10   ⎜  1   ╲╱ 3 ⋅ⅈ⎟    ╱  2⋅╲╱ 11523    2467                        133     \n",
       "⎪── + ⎜- ─ - ───────⎟⋅3 ╱   ─────────── + ────  + ────────────────────────────\n",
       "⎨3    ⎝  2      2   ⎠ ╲╱         3         27                            _____\n",
       "⎪                                                   ⎛        ___  ⎞     ╱     \n",
       "⎪                                                   ⎜  1   ╲╱ 3 ⋅ⅈ⎟    ╱  2⋅╲╱\n",
       "⎪                                                 9⋅⎜- ─ - ───────⎟⋅3 ╱   ────\n",
       "⎩                                                   ⎝  2      2   ⎠ ╲╱        \n",
       "\n",
       "                                                                              \n",
       "                                                                       ⎛      \n",
       "                    10                       133                       ⎜  1   \n",
       "───────────────: 1, ── + ─────────────────────────────────────────── + ⎜- ─ + \n",
       "_______________     3                           ____________________   ⎝  2   \n",
       "_______                    ⎛        ___  ⎞     ╱     _______                  \n",
       " 11523    2467             ⎜  1   ╲╱ 3 ⋅ⅈ⎟    ╱  2⋅╲╱ 11523    2467           \n",
       "─────── + ────           9⋅⎜- ─ + ───────⎟⋅3 ╱   ─────────── + ────           \n",
       " 3         27              ⎝  2      2   ⎠ ╲╱         3         27            \n",
       "\n",
       "              ____________________                                            \n",
       "  ___  ⎞     ╱     _______                                                    \n",
       "╲╱ 3 ⋅ⅈ⎟    ╱  2⋅╲╱ 11523    2467                  133               10      ╱\n",
       "───────⎟⋅3 ╱   ─────────── + ──── : 1, ─────────────────────────── + ── + 3 ╱ \n",
       "   2   ⎠ ╲╱         3         27              ____________________   3    ╲╱  \n",
       "                                             ╱     _______                    \n",
       "                                            ╱  2⋅╲╱ 11523    2467             \n",
       "                                       9⋅3 ╱   ─────────── + ────             \n",
       "                                         ╲╱         3         27              \n",
       "\n",
       " ____________________   ⎫\n",
       "╱     _______           ⎪\n",
       "  2⋅╲╱ 11523    2467    ⎪\n",
       "  ─────────── + ──── : 1⎪\n",
       "       3         27     ⎬\n",
       "                        ⎪\n",
       "                        ⎪\n",
       "                        ⎪\n",
       "                        ⎭"
      ]
     },
     "execution_count": 124,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.eigenvals()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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