Intro_Ipython_Numpy_Scipy_solution.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Ipython notebook\n",
    "\n",
    "Ipython est initialement un projet de console python spécialisé pour les applications scientifiques.\n",
    "Avec le temps, le projet s'est élargi et inclus désormais un ensemble d'utilitaires pour les applications scientifiques, dont le notebook.\n",
    "\n",
    "Ipython notebook est une feuille de calul interactive, inspirée de Maple et de Mathematica, qui fonctionne en mode web service. Au démarrage d'un notebook, Ipython lance un serveur d'application dans lequel tourne un kernel (noyau d'application) Ipython et le navigateur par défaut de votre OS sur l'adresse localhost 127.0.0.1. \n",
    "\n",
    "Ce mode de fonctionnement client/serveur permet notamment de lancer des calculs en se connectant à un supercalculateur ou encore à un data center de type Amazon Web Service, Microsoft Azure ou Google App Engine.\n",
    "\n",
    "\n",
    "Le notebook est un environnement de calcul interactif complémentaire avec Spyder.\n",
    "En tant qu'IDE, Spyder est plus adapté au développement de fonctions et d'applications.\n",
    "Le notebook est plus adapté à la réalisation d'étude ou au traitement de donnée."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Fonctionnement de l'Ipython Notebook\n",
    "\n",
    "Dans les versions >2.x de Ipython, le notebook s'utilise en basculant constemment entre 2 modes : \n",
    "\n",
    "* mode commande des cellules: \n",
    "    * Accessible en appuyant sur ESC\n",
    "    * Permet de créer (A et B), supprimer (X), copier (C) et coller (V) des cellules\n",
    "    * Changer de type de cellule : code (Y) ou markdown (M)\n",
    "\n",
    "* mode édition\n",
    "    * Accessible en appuyant sur ENTER\n",
    "    * Executer le code de la cellule (SHIFT ENTER) ou afficher le rendu d'une cellule mardown\n",
    "\n",
    "L'ensemble des commandes est accessible dans la barre d'outil (edit). L'ensemble des raccourcis est disponible dans le menu help de l'interface (help -> keyboard shortcut), ainsi qu'une visite guidée de l'interface (help -> user interface tour)\n",
    "\n",
    "Pour plus d'éléments, se rérérer à la documentation officielle : \n",
    "\n",
    "http://ipython.org/ipython-doc/2/notebook/notebook.html#structure-of-a-notebook-document\n",
    "\n",
    "Une galerie de notebook est disponible en ligne via l'application web nbviewer qui permet la visualisation de fichiers .ipynb hébergés sur Github : \n",
    "http://nbviewer.ipython.org/"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Numpy, Scipy et Matplotlib\n",
    "\n",
    "Numpy, Scipy et Matplotlib sont les bibliothèques de calcul numérique les plus importantes de Python.\n",
    "\n",
    "## Numpy \n",
    "\n",
    "Inclus : \n",
    "\n",
    "* l'ensemble des fonctions mathématiques\n",
    "* les manipulations sur les vecteurs et les matrices\n",
    "http://wiki.scipy.org/NumPy_for_Matlab_Users\n",
    "\n",
    "Il comprend approximativement l'ensemble des fonctionalités de base de Matlab "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 2]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np # np est une convention d'import\n",
    "print np.array([0,1,2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Le numpy array est l'équivalent du vecteur matlab : "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 2 3 4 5 6 7 8 9]\n"
     ]
    }
   ],
   "source": [
    "x = np.arange(0,10)\n",
    "print x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0  1  4  9 16 25 36 49 64 81]\n",
      "[ 0  2  4  6  8 10 12 14 16 18]\n",
      "285\n"
     ]
    }
   ],
   "source": [
    "print x**2 # mise à la puissance 2. Toute les opérations sont par défaut élément par élément (elemnt wise)\n",
    "print x+x\n",
    "print x.dot(x) # produit scalaire"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0 1]\n",
      " [1 0]]\n"
     ]
    }
   ],
   "source": [
    "x2 = np.array([[0,1],[1,0]]) # matrice\n",
    "print x2 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 6 5]\n",
      " [7 3 8]\n",
      " [8 7 2]]\n",
      "[[1 7 8]\n",
      " [6 3 7]\n",
      " [5 8 2]]\n",
      "[[ 2 18 20]\n",
      " [14  9 32]\n",
      " [16 21  8]]\n",
      "[40 55 45]\n"
     ]
    }
   ],
   "source": [
    "x3 = np.random.randint(low=0,high=10,size=[3,3])\n",
    "u3 = np.array([2,3,4])\n",
    "print x3\n",
    "print x3.T # transposée\n",
    "print x3*u3# multiplication de chaque ligne par le vecteur u3\n",
    "print x3.dot(u3) # produit matricielle"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "De manière générale, les manipulations purement matricielles sont souvent plus lourdes en termes de syntaxe que Matlab&reg;  (MATrix LABoratory) qui est spécialisé sur ce type d'objet. Cependant, les performances sont similaires. \n",
    "\n",
    "Un benchmark sur ce type de manipulation réalisé par la NASA entre Matlab, Python, Java et Fortran est disponible en ligne : https://modelingguru.nasa.gov/docs/DOC-1762 .\n",
    "\n",
    "En substance matlab et numpy ont des performances similaires, Fortran (avec compilateur intel) est toujours meilleur.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matplotlib \n",
    "\n",
    "La bibliothèque graphique la plus populaire pour les graphes 2D (Mayavi pour la 3D).\n",
    "La galerie en ligne est munie de nombreux exemple de codes : \n",
    "\n",
    "http://matplotlib.org/gallery.html\n",
    "\n",
    "Un tutorial de matplotlib : http://nbviewer.ipython.org/github/cmiller8/PythonforBuildingAnalysts/blob/master/0_PythonBaseLibraries/3_MatplotlibLibrary.ipynb"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import matplotlib.pylab as plt\n",
    "import seaborn as sns\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La ligne précédente est une commande spécifique à Ipython. Elle ne fait pas partie du langage Python.\n",
    "Les commandes commançant par **%** ou **%%** sont des **magic** qui configurent le comportement du notebook.\n",
    "**%matplotlib inline**  indique que les graphes seront inclus en ligne dans le notebook (et sauvegardés avec lui).\n",
    "Les graphes interactifs s'obtiennent avec la commande **%matplotlib qt **"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x18f2fda0>]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
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       "5MmNKqX0ZS7+QwNl3Ltu1wf537vTKaVTETFLJ4QfqxzO/v71uD7I/P4BpJR+GxEHgH8DvlA5NND9\n",
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       "J53n7vsjYvPKuwe6f3U9c6/1NWkaYMPri4htwBMR8Wo6z8VuBu6bylmOXwn3bkMl3LuIeDnwKPD+\n",
       "lNLhNYezv3/drq+Q+3cn8EcppX8CzgAv0PnCKgx4/+qK+1eAWyLiyMrb74qIPZx/TZoPA9/g/GvS\n",
       "/Lqm86hLr+vbCxym850030wpPTKtEx3RMkBh965qvevL/d7to/NP9bsjYvXZ9H5gSyH3r9f15X7/\n",
       "/hM4EBHfAjYBHwR2R8TAf/58bRlJKpA/xCRJBTLuklQg4y5JBTLuklQg4y5JBTLuklQg4y5JBTLu\n",
       "klSg/weFbQmfIo3R0gAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18ddfb38>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot([0,1,2,3],[0,2,1,2],'s-')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Création de 100 points de 0 à $2\\pi$ dans le vecteur $x_4$, puis calcul de \n",
    "$$y_4 = \\sin(x4)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18e5b780>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x4 = np.linspace(0,5*np.pi,100) \n",
    "y4 = np.sin(x4)\n",
    "\n",
    "fig = plt.figure() # création d'une fenetre de graph\n",
    "ax1 = fig.add_subplot(121) # graphe de gauche sur 2 graphes répartis horizontalement\n",
    "ax2 = fig.add_subplot(122) # graphe de droite sur 2 graphes répartis horizontalement \n",
    "ax1.plot(x4,y4)\n",
    "ax2.plot(y4,x4)\n",
    "ax1.set_title(r'$sin(x)$')\n",
    "fig.savefig('test.png',dpi=150)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Le code suivant génère un échantillon de taille 1000 tiré dans la loi normal multivariée : \n",
    "\n",
    "$$x_5 \\sim  \\mathcal{N}(\\mu,\\Sigma)$$\n",
    "Avec $$\\mu= \\left(\\begin{array}{cccc} 20 \\\\ 30 \\\\ 100\\end{array} \\right) $$\n",
    "\n",
    "$$\\Sigma= \\left(\\begin{array}{cccc} \\\\10&-5&0.3\\\\-5&10&0.2\\\\0.3&0.2&1000\\end{array} \\right)$$\n",
    "\n",
    "Puis on trace uniquement les 2 premières composantes du vecteur dans un graphe de dispersion (*scatter*). On utilise la dernière composante pour la taille et la couleur des points."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x1a6301d0>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       "usfj8awI3qB7PB7PiuANusfj8awI3qB7PB7PiuANusfj8awI3qB7PB7PiuANusfj8awI/w8tzMGI\n",
       "W73vrgAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x19336e10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x5 = np.random.multivariate_normal(mean=[20,30,100],cov=[[10,-5,0.3],[-5,10,0.2],[0.3,0.2,1000]],size=1000)\n",
    "plt.scatter(x5[:,0],x5[:,1],s=x5[:,2],c=x5[:,2],edgecolor='none',alpha=0.15,cmap='hot')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "A1+SCmHgS1IhDHxJKoSBL0mFMPAlqRAGviQV4v8BVWiQfj5C8w0AAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18eb5978>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "res = plt.hist(x5.flatten(),bins=50)  # que fait la méthode flatten ? "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Scipy \n",
    "\n",
    "Comprend l'équivalent des principales toolbox de Matlab."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from scipy import linalg, optimize, sparse # par exemple : algèbre linéaire, optimisation et traitement de matrice crzeuse\n",
    "from copy import copy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# TP : manipulation matricielle"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "http://en.wikibooks.org/wiki/LaTeX/Mathematics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercice :  \n",
    "* En utilisant la fonction diag de numpy, écrire la fonction permettant de construire la matrice tridiagonale de taille $N_x$ suivante:\n",
    "\n",
    "$$ A = \\left( \\begin{array}{cccc}\n",
    "1+2Fo & -Fo & & 0\\\\\n",
    "-Fo & 1+2Fo & \\ddots & \\\\\n",
    "\\\\\n",
    "& \\ddots & 1+2Fo & -Fo \\\\\n",
    "0 & & -Fo & 1+2Fo \\end{array} \\right)$$\n",
    "\n",
    "Où \n",
    "\n",
    "$$Fo = \\frac{\\alpha \\Delta t}{\\Delta x^2}$$\n",
    "\n",
    "Et le vecteur :\n",
    "\n",
    "$$b^{i-1} = \\left(\\begin{array}{cccc} \\theta^{i-1}_1+Fo\\theta^{i-1}_{0} \\\\ \\vdots \\\\ \\theta^{i-1}_k\\\\ \\vdots \\\\ \\theta^{i-1}_{N_x}+Fo\\theta^{i-1}_{N_x+1} \\end{array} \\right) $$\n",
    "\n",
    "\n",
    "* En utilisant la fonction solve du package scipy.linalg, trouvez $T^i$ tel que : \n",
    "\n",
    "$$ AT^i = b^{i-1} + \\frac{\\Delta t}{\\rho C_p}S^{i-1}$$\n",
    "\n",
    "Où $S$ est un terme source en $W.m^{-3}.$\n",
    "\n",
    "** NB : Il s'agit de la résolution de l'équation de la chaleur en 1D en instationnaire sur un schema centré implicite.**\n",
    "\n",
    "**$\\theta^{i-1}_{0}$ et $\\theta^{i-1}_{n+1}$ sont les conditions limites à gauche et à droites**\n",
    "\n",
    "**Fo est le nombre de Fourier de maille. Sur ce schéma numérique implicite, la solution est inconditionnellement stable, mais la précision est dégradée pour Fo > 1.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercice : **\n",
    "\n",
    "* Caluler l'évolution de la température pour $N_t$ itérations temporelles et $N_x$ mailles spatiales. \n",
    "* Stocker le résultat dans une matrice, soit en affectant les valeurs dans une matrice définie à l'avance, soit en concatenant les vecteurs avec la fonction np.hstack\n",
    "* Utilisez la commande **%%timeit** pour vérifier la performance de votre code\n",
    " "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Définition des paramètres physiques du système**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "alpha = 0.54e-6  # m2.s-1\n",
    "rho   = 2.4e3    # kg.m-3\n",
    "Cp    = 0.88e3   #J.kg-1.K-1 \n",
    "\n",
    "dt = 10*60. # s\n",
    "dx = 0.01 # 1 m\n",
    "\n",
    "L = 0.5 # 2 m \n",
    "duration = 3*24*3600. # 3 jours\n",
    "\n",
    "Nx = int(L/dx)\n",
    "Nt = int(duration/dt)\n",
    "\n",
    "Fo = alpha*dt/(dx**2) # Fourier de maille. Peremt d'apprecier la précision et la stabilité du schéma numérique"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Définition de la matrice $A$**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "A = np.diag(-Fo*np.ones(Nx-1),-1)+\\\n",
    "    np.diag(1+2*Fo*np.ones(Nx))+\\\n",
    "    np.diag(-Fo*np.ones(Nx-1),1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 7.48 -3.24  0.   ...,  0.    0.    0.  ]\n",
      " [-3.24  7.48 -3.24 ...,  0.    0.    0.  ]\n",
      " [ 0.   -3.24  7.48 ...,  0.    0.    0.  ]\n",
      " ..., \n",
      " [ 0.    0.    0.   ...,  7.48 -3.24  0.  ]\n",
      " [ 0.    0.    0.   ..., -3.24  7.48 -3.24]\n",
      " [ 0.    0.    0.   ...,  0.   -3.24  7.48]]\n"
     ]
    }
   ],
   "source": [
    "print A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Définition du vecteur $b$**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def rhs(Ti,Tg,Td,Fo):\n",
    "    \"\"\"\n",
    "    rhs : right hand side of the equation\n",
    "    \n",
    "    Calcul le vecteur b tel que Ax = b \n",
    "    \n",
    "    Ti vecteur de solution à l'instant i\n",
    "    Tg condition limite à gauche en température (scalaire)\n",
    "    Td condition limite à droite en température (scalaire)\n",
    "    Fo : Fourier de maille\n",
    "    \"\"\"\n",
    "    b = copy(Ti)\n",
    "    b[0]  += Fo*Tg \n",
    "    b[-1] += Fo*Td \n",
    "    return b    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Résolution du système pour $N_t$ pas de temps**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "#%%timeit\n",
    "Ti = 15*np.ones([Nx,1])          # Condition initiale à 15°C\n",
    "t = np.linspace(0,duration,Nt)   # vecteur temps\n",
    "Tg = 10 + 2*np.sin(t/(24*3600)*2*np.pi)  # Condition limité à gauche sinusoidale centré autour de 10°C\n",
    "TT = []                         # liste vide dans laquelle on va ajouter les profiles de température à chaque pas de temps\n",
    "TT.append([Ti])\n",
    "\n",
    "for t in range(Nt):   \n",
    "    b = rhs(Ti,Tg[t],19,Fo)\n",
    "    Ti = np.linalg.solve(A,b)    # Résolution d'un système d'équation linéaire AX=b\n",
    "    TT.append([Ti])\n",
    "\n",
    "T =  np.vstack(TT)\n",
    "T = np.reshape(T,[Nt+1,Nx])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercice : \n",
    "\n",
    "* Tracez de manière superposée le profil de température $\\theta$ pour les 10 premiers pas de temps.\n",
    "\n",
    "* Représentez les courbes d'isotempératures annotée en noire sur le champs de température représenté par la colormap \"hot\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "TkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1a86de48>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "with sns.color_palette(\"Blues_r\",12): # définition du cycle des couleurs des courbes\n",
    "    ax = plt.plot(T[0:10,:].T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "X,Y  = np.meshgrid(np.arange(Nx),np.arange(Nt+1)) # Construction des matrices X et Y pour la fonction countour"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0  1  2 ..., 47 48 49]\n",
      " [ 0  1  2 ..., 47 48 49]\n",
      " [ 0  1  2 ..., 47 48 49]\n",
      " ..., \n",
      " [ 0  1  2 ..., 47 48 49]\n",
      " [ 0  1  2 ..., 47 48 49]\n",
      " [ 0  1  2 ..., 47 48 49]]\n",
      "\n",
      "[[  0   0   0 ...,   0   0   0]\n",
      " [  1   1   1 ...,   1   1   1]\n",
      " [  2   2   2 ...,   2   2   2]\n",
      " ..., \n",
      " [430 430 430 ..., 430 430 430]\n",
      " [431 431 431 ..., 431 431 431]\n",
      " [432 432 432 ..., 432 432 432]]\n"
     ]
    }
   ],
   "source": [
    "print X\n",
    "print \n",
    "print Y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Anaconda\\lib\\site-packages\\matplotlib\\text.py:52: UnicodeWarning: Unicode equal comparison failed to convert both arguments to Unicode - interpreting them as being unequal\n",
      "  if rotation in ('horizontal', None):\n",
      "C:\\Anaconda\\lib\\site-packages\\matplotlib\\text.py:54: UnicodeWarning: Unicode equal comparison failed to convert both arguments to Unicode - interpreting them as being unequal\n",
      "  elif rotation == 'vertical':\n"
     ]
    },
    {
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       "AAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1aa6f7f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = plt.imshow(T.T,cmap='RdBu_r',aspect='auto',interpolation='nearest') # création de la carte des températures\n",
    "\n",
    "# Cration des contours\n",
    "CS = plt.contour(Y,X,T,10,\n",
    "                 colors='k',linewidth=1)   # negative contours will be dashed by default\n",
    "                 \n",
    "plt.clabel(CS, fontsize=10, inline=0.5)    # tracer des labels\n",
    "\n",
    "f = ax.get_figure()                        # récupération de l'objet figure qui \"encapsule\" l'objet ax\n",
    "\n",
    "f.savefig('Chronogram_Temperature_1D.png',dpi=150) # Sauvegarde du graphe"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sauvegarder vos résultats : \n",
    "\n",
    "Sauvez la matrice T au format csv avec la fonction savetxt de numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "np.savetxt('temperature_1D_implicit.csv',T,delimiter=';')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bonus : Calcul symbolique avec Sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing() # formatte les sorties avec Latex"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "X = Symbol('X')\n",
    "x, y, z, t, i = symbols('x y z t i')\n",
    "k, m, n = symbols('k m n', integer=True)\n",
    "f, g, h = symbols('f g h', cls=Function)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": [
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       "2I41uB578F6L9Xe0wDim+u7vwG9YtzxyOv6LLTiJjfn9asyJDus4hRgRU18xlV4uOmpi2RT9T5jB\n",
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       "F/FIzbbBGXhZLMoF23IT76RU1IBMvVG5Gr+LSK1/0zqmd+pQFX0O02Mm/Yvaa/5ebsEtwqgITat2\n",
       "9B+JEXppoohByNQzuwp/4guc05d+tXjpH1SUGbbHTFpH3av6S3qu+FOOitaLnXLxmZaZFp/o7gQR\n",
       "w+YPPCZOIGb70u8Xi3JZ66nm8RahfwNe6LtmVOyjJnFETB/FIeQNpTzT+EVv3j9iuGYyaaPyHrG5\n",
       "PSgivkNiI1mwlB4z9bXP6k3R5Wspo+yByTSL+paTTMvah3kq3JTjYlpbiaxk7R0dHR0dHUn8A979\n",
       "63P+p0WJAAAAAElFTkSuQmCC\n"
      ],
      "text/latex": [
       "$$X^{2} + 3 X + 2$$"
      ],
      "text/plain": [
       " 2          \n",
       "X  + 3⋅X + 2"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((X+1)*(X+2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "/TzNxDeRncdEdbgm4R3swfNN7plygeX4oEmjJ4We7o98tzSJL1JNZ1aJhJ2DByN7NCa6y/Uynkt8\n",
       "a3Li6lxx+TgofPwN6yq8Jqy0YIWwNXQusydjc4sJdlqL8Vlkp0xUh2utMIduSfzLcmLrXHFJPI9j\n",
       "Qon4WtgS2YzDwtLyVo0T+KBQHiZKw/uA/UYebqyFRnIdVspENbgGhHl0D7ZH/ikurSJ5XHUtwbbs\n",
       "+Dqhvm4V3sjpAsibWcyKMSZbU750zMbHOCqAXMRPwnyzOondJmxlpYqZqAbXkBGe9JeOuCKuul4R\n",
       "hmCnVDO2OayZFgnzQZE6zUR3uBBWLoM691fEGxq/lcarK4WJO92SitVpJrrD1VNPPfX0f9Z/f/XP\n",
       "zzBcwxwAAAAASUVORK5CYII=\n"
      ],
      "text/latex": [
       "$$\\left(x + 1\\right) \\left(x + 2\\right)$$"
      ],
      "text/plain": [
       "(x + 1)⋅(x + 2)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(x**2+3*x+2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "jfsN2grTaFtMYyQawGT+islwtNn1USrLmlcgN8Cxji8LsC/vn4eMaQ7yz89HQr8C9SLqYQRdzxRM\n",
       "8grZZ0cme8G1CHBzS+2BVnJdjdwjF1CpxKF7ymcwKjPRMVWrUOM/xSFdAW5y3CB85nq00XshXd0y\n",
       "lvLdDvKOmILmqiZkfO+il53QCOZStBn9Bqq3JbkQYNeNa9mplWdLuEbq0Kq/PZE9L0YjuutJPyHL\n",
       "qhdL+Y4ow/7A3QY7UGP+cPjO11Ej9xXs25tA+bqIxttiFpejudamEND4A8UtnIkOxG0WAe2RbyvN\n",
       "lo+VgO4lRyvtKm8XAnqmbiz0BP1ZCVh79dyFRp3M8xq15xmaweHAR5r4vnbJt5Vmy8dKd5OjlXaV\n",
       "tws9VTcWeoL+rKzNel6rGEv1fiZPNV4+zcXLu3vj9bcW0+pVr40k7QdVPRW8fJqLl3f3xuvP0yO5\n",
       "merfCfRU4+XTXLy8uzdefx6Px+PxeDwej8fj8Xg8Ho/H4/F4PB6Px+NpCv8HSRZQKx56En4AAAAA\n",
       "SUVORK5CYII=\n"
      ],
      "text/latex": [
       "$$1 + \\frac{x^{2}}{2} + \\frac{5 x^{4}}{24} + \\frac{61 x^{6}}{720} + \\frac{277 x^{8}}{8064} + \\mathcal{O}\\left(x^{10}\\right)$$"
      ],
      "text/plain": [
       "     2      4       6        8         \n",
       "    x    5⋅x    61⋅x    277⋅x     ⎛ 10⎞\n",
       "1 + ── + ──── + ───── + ────── + O⎝x  ⎠\n",
       "    2     24     720     8064          "
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(1/cos(x)).series(x, 0, 10) # développement en série limitée autour de 0 à l'ordre 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": [
       "iVBORw0KGgoAAAANSUhEUgAAAAsAAAASCAYAAACNdSR1AAAABHNCSVQICAgIfAhkiAAAAMNJREFU\n",
       "KJHF0D1KA1EUhuHHIIhMCisTTJNiNpAUFmKVXaQJZCE2apEqO0gbxMJOIjaCpa0LCFhKYBobC3+K\n",
       "mYHJYWCKFPngFvfl5X7nHnbIOR6wwjsW6NWJQzzjpLi38YpP9KP8iDSwAf5wF+UvfOA08AwbaFXg\n",
       "Gh0kQf7GcXw5QTews2KMlyjXZYYfXDSJqfwft03iEd4wbxIPsMR1k6iovQpswvbqYIpf3AR+WVaW\n",
       "GeEeT0E8lK9vXIVZAetObNpn/gF5aCRCZZOlJQAAAABJRU5ErkJggg==\n"
      ],
      "text/latex": [
       "$$2$$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "summation(1/2**i, (i, 0, oo)) # somme infinie de 1/2^n "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": [
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       "qFTeCDHn9B3mR8IjvJjT9sFi4fMuFs4QQu+3tg54lrCsEjOUP6zTTNTsGnbBgjJlu4nY6dYq2p0m\n",
       "lGC1cdXjQur/U2aLGCnLZLGRuqfeTcfyzNhG4sKCtudhh5z3/cSZmaWspZW4Aa+LxdpIBHcrCgaQ\n",
       "5bTkj5DSfcqU7S0s67Mq2h0pFqHaa6+d1HY9tJnYJFlG42WsTr3bWyjV9Ni+FR4qS09sJeLGLKO0\n",
       "vtYqCZpxwnvMwSNFg35KiIlpQpxUw97C1ZXOs/FCTRZJ5q5i116ckzck6Z9QrKu1VHLl6CU+4IQq\n",
       "y+fxnPwNvUgo1xLdxRyzIm2QUMNZJii2wDwGyLG0ouD6euHm/sB3VXZwnghg/0zSy8SCjSooX3KN\n",
       "Y7R0LfthCu5N0iPEwfxKleMoXQB/UGX5PL4VVpFlqZab+Eoxx6wr7infUsfJF3pFdMv8r8ibKguJ\n",
       "NEvEIV2iSUzy0DJ1uopL2wdwDW4UIUB6EY8XNxrVnmdThHBq6xVdmhniYjzLYHF0zBShyPykr+zm\n",
       "P1RrS++n+nO2SbjhZcLSVon1OLzK+msdc/FwjW2MxOkVynTGV1qq5xJXaB1unIuzahzXf4oLhGsm\n",
       "5Pn4GtvrqnXgPlfLoPoIIc2HZMp1kn+eLVBemP3vWCjE01AhFtrjp6NxImwpsVyzJO+PxeLSN8th\n",
       "Wgf9pd/12oUulYusFawU4cn+wgV93Q5tLhRXcIuEaFoszvgxwsqmiiA5TV9sJ67v0kwS51/2+qvB\n",
       "v5gnFF8DNmjQoEGDBv9b/gYDoxdsCZK7UgAAAABJRU5ErkJggg==\n"
      ],
      "text/latex": [
       "$$\\frac{\\sqrt{\\pi}}{4} \\operatorname{erf}^{2}{\\left (y \\right )}$$"
      ],
      "text/plain": [
       "  ___    2   \n",
       "╲╱ π ⋅erf (y)\n",
       "─────────────\n",
       "      4      "
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(exp(-y**2)*erf(y), y) # intégration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": [
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       "+BoTcbtwD8c6o/EotmEMrlfegrMSjcaItcDNeBn/YXLZjfaIl3B3Sk/Bv0K4XlOZRstwUCzcFqJv\n",
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       "/4EC1uFyEUMbm/KLsbmHbRRpdJvo9Fmdeq7RWcKXLs+UXYcTc3We1xiB/fgOlwlxDuC+9N2qThrN\n",
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       "H8jUn6u8afmIo7hG4U6sTlerOq1opRHNOh2VRkUP54L0mW94O8al9NXph2djFmZib6buw8JdDTAL\n",
       "nxa0VyXXiN6/Bhfr/ExBkUY0dDoqjYoezoXC/+UbfgVLUvpD4XYWiLjSJWJ4voBn8XG6CF+7Sfnu\n",
       "pxu6PVNQpBENnUrRaJ04mVI1/cpf53R7pqB0jbIjZzXeS+mLsL7MhjtkJ/aU3MZe8RJehA/wS5u6\n",
       "lWn0Fd4UEd1vxBbuUCa7LX04Oj1TUJlGK/EiXsV5g9VoiQxsS3dCp2cKVhpeGlVGp9vSx+yZgqI5\n",
       "/FAkvy29QkyP3xABxpli0bhZxLeuEqGTH6ow9nijaFt6oka091p8ktJrFYeTanpIu23pcWIkwRMi\n",
       "QjwkGC4H2ZeKh7E75ZfgXRGE3KMROO3DhpSeNJgGdsNweTjttqWX4R6xmJ2Dz8W7duVgG3mkDJcJ\n",
       "wTgxM9sgVvxTxX/bjFNE+GSrcHv7xWhaj9+qMLampqampqampmP+B912CFXfAymYAAAAAElFTkSu\n",
       "QmCC\n"
      ],
      "text/latex": [
       "$$f{\\left (x \\right )} + \\frac{d^{2}}{d x^{2}}  f{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "         2      \n",
       "        d       \n",
       "f(x) + ───(f(x))\n",
       "         2      \n",
       "       dx       "
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = Function('f')\n",
    "f(x).diff(x, x) + f(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "+9z0wysZt52iWIfjx1l6m2b/o4jj8JyIl9exSDg+xBX3ZJu612vYl2VxMX6bpW/Fe7E3nupQ75Mm\n",
       "nvj7ioeuizScs+V4QjhyxFX8RWGbb9R98ZdBJ0774bOd31DD50z0g1phsnweJWz7Vwr7/gv4nVjU\n",
       "dA4X/yt9H9CirG4CPar73PTDK525RcS/x8ULeo7ckdlL/O3gTSm9UXNIeKb4a0od7xLkDQsrhbN1\n",
       "fgeZk/HpQt45YtHkpg9hDl1WkH2FIP/ALG+z9iZWMX9Ni37qnI6Cz5ryJtZKvfM5G1/J0qeKzXWw\n",
       "OIwf1BwBrWOViPrNw//w+RYyG1L93ZSbm8nySglu361h7+U4U4TU4ANJ5nTxkvo33JLJXi7CcHWc\n",
       "q9lhHQYuETt/tWbzcUycZJ1MgVYbZKuwd+t4PbYU6v0B32vR3sMt8j+a+slt9jqno+CzpreHwl75\n",
       "XCo21aKUniPGdGpKnyBCuycX2vp2lj5DmGEHZXkLhPN+bEqv0X1uJssrGbftCL5U7ND9NduDs9Ig\n",
       "N6WyryfFx8Wuv0YQ8Bx+gl+kekuwu4g8DBtHCwetTuoOYTZdKyIz7TCe6tyf0qsF6fsLh/UlETP/\n",
       "qjjJjhHRj7cI+/gWEaI8XpB+ZMrfIkKO61LZ3NTHDcJhrXN6m+HzWVPexKqjFz5niMflLWkMhyUd\n",
       "j8RfksyKpMM2YVI9KzjdnrXzTnwsybwozLxvicVN97lhcrxSktub8aNOArsgijfIroia0Y7xJvGY\n",
       "Om2RX5uf0dhJy/HD0atTYcjYIU7OUeBscUNcMqL+ho47hOm0VDxw7T616owcL4cbZFR4n8Zr9J6m\n",
       "Ma/5DXKFcCwvE47N8y1rVKjQGceISNIm8ZZwomaHu8I0wxkiijIuHp4u6CxeoQMWiqjXeOEzZyqV\n",
       "qlChQoUKFSpUqFChQoXpgv8DR63P/6GqB9oAAAAASUVORK5CYII=\n"
      ],
      "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": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsolve(f(x).diff(x, x) + f(x), f(x))   # résolution d'équations différentielles"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Anaconda\\lib\\site-packages\\IPython\\core\\formatters.py:239: FormatterWarning: Exception in image/png formatter: \n",
      "\\begin{Bmatrix}x : -3, & y : 1\\end{Bmatrix}\n",
      "^\n",
      "Unknown symbol: \\begin (at char 0), (line:1, col:1)\n",
      "  FormatterWarning,\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$$\\begin{Bmatrix}x : -3, & y : 1\\end{Bmatrix}$$"
      ],
      "text/plain": [
       "{x: -3, y: 1}"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve([x + 5*y - 2, -3*x + 6*y - 15], [x, y]) # résolution d'équation linéaire"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sympy version   \t:\t 0.7.5\n",
      "IPython version \t:\t 2.4.1\n",
      "numpy version   \t:\t 1.9.2\n",
      "statsmodels version \t:\t 0.5.0\n",
      "matplotlib version \t:\t 1.9.2\n"
     ]
    }
   ],
   "source": [
    "import IPython\n",
    "import sympy\n",
    "from statsmodels import version as sm_version\n",
    "print \"Sympy version   \\t:\\t %s\"%sympy.__version__\n",
    "print \"IPython version \\t:\\t %s\"%IPython.__version__\n",
    "print \"numpy version   \\t:\\t %s\"%np.__version__\n",
    "print \"statsmodels version \\t:\\t %s\"%sm_version.full_version\n",
    "print \"matplotlib version \\t:\\t %s\"%plt.__version__"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}