NumPy - Applications

numpy-applications.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# NumPy : applications"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dérivées numériques"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pour un traitement numérique, une fonction $f(x)$ est souvent représentée par ses valeurs sur une grille $x_i$, $i = 1\\ldots N$. Nous en avons vu en exemple la semaine dernière, en plottant les fonctions $\\sin$ et $\\cos$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = np.linspace(0, 2.*np.pi, 50)\n",
    "sin_x = np.sin(x)\n",
    "cos_x = np.cos(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercice : dérivée numérique"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Étant donné $x_i$ et $f(x_i)$ pour $i=1\\ldots N$, l'approximation la plus simple pour la dérivée $f'$ est\n",
    "\n",
    "$$x'_j = \\frac{1}{2}(x_j + x_{j+1})$$\n",
    "\n",
    "$$f'(x'_j) = \\frac{f(x_{j+1})-f(x_j)}{x_{j+1}-x_{j}}$$\n",
    "\n",
    "pour $j = 1 \\ldots N-1$.\n",
    "\n",
    "Calculez cette dérivée numérique pour la fonction $\\sin$ et faites un plot comparatif avec la fonction $\\cos$.\n",
    "\n",
    "**Pour bien profiter des avantages en performance de NumPy, il est important de ne pas utiliser des boucles explicites dans votre code. Le `for` est donc interdit dans cet exercice !**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "hide_input": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x11366bba8>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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6BdgR9royWBbPNvHsKyIiaZQzjc9mNtHMKsysoqqqKtPJKQxBADPFUJJki8hTCpaXVZJR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\ngkFERGr5/8oa2oLoftJlAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11340c160>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xx = 0.5*(x[1:]+x[:-1])\n",
    "ff = (sin_x[1:]-sin_x[:-1]) / (x[1:]-x[:-1])\n",
    "plt.plot(xx, ff, '*', label=\"dérivée de sin\")\n",
    "plt.plot(x, cos_x, '+', label=\"cos\")\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Les matrices sont des tableaux avec deux dimensions. Elles sont fréquemment utilisées en calcul scientifique, souvent dans les applications de l'algèbre linéaire (systèmes d'équations linéaires, équations différentielles, opérateurs linéaires, ...), mais aussi pour des applications très différentes comme la représentation d'un graphe (matrices d'adjacence).\n",
    "\n",
    "Deux opérations fréquentes sont la multiplication de deux matrices :\n",
    "\n",
    "  $$A_{ij} = \\sum_{k} B_{ik} \\cdot C_{kj}$$\n",
    "\n",
    "et la multiplication d'un vecteur (tableau 1D) avec une matrice :\n",
    "\n",
    "  $$y_{i} = \\sum_{k} M_{ik} \\cdot x_{k}$$\n",
    "\n",
    "Les deux sont écrites de la même façon en NumPy :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "vecteur = np.array([1, 2])\n",
    "matrice1 = np.array([[10, 20], [30, 40]])\n",
    "matrice2 = matrice1.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Le suffix `.T` produit la transposée de la matrice d'origine."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 500, 1100],\n",
       "       [1100, 2500]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrice1 @ matrice2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 50, 110])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrice1 @ vecteur"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Il ne faut surtout pas confondre la multiplication de matrices avec la multiplication élément par élément de deux tableaux :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 100,  600],\n",
       "       [ 600, 1600]])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrice1 * matrice2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercice : géométrie 2D"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Un point dans un plan (papier, écran, ...) peut être représenté par un vecteur avec deux éléments $r = [x, y]$. Les deux transformations les plus basiques dans un plan sont la translation,\n",
    "\n",
    "   $$r' = r + [\\Delta_x, \\Delta_y]$$\n",
    "\n",
    "et la rotation autour de l'origine par un angle $\\phi$,\n",
    "\n",
    "   $$r'' = D(\\phi) \\cdot r$$\n",
    "\n",
    "avec la matrice de rotation\n",
    "\n",
    "$$\n",
    "   D(\\phi) =\n",
    "     \\left( \\begin{matrix}\n",
    "       \\cos\\phi & -\\sin\\phi \\\\\n",
    "       \\sin\\phi & \\cos\\phi \\\\\n",
    "     \\end{matrix} \\right)\n",
    "$$\n",
    "\n",
    "En rajoutant un troisième élément $1$ aux vecteurs $r = [x, y, 1]$, on peut écrire une rotation suivie par une translation comme\n",
    "\n",
    "   $$ r''' = T(\\phi, \\Delta) \\cdot r$$\n",
    "\n",
    "avec\n",
    "\n",
    "$$\n",
    "   T(\\phi, \\Delta) =\n",
    "     \\left( \\begin{matrix}\n",
    "       \\cos\\phi & -\\sin\\phi & \\Delta_x \\\\\n",
    "       \\sin\\phi & \\cos\\phi  & \\Delta_y \\\\\n",
    "       0 & 0 & 1 \\\\\n",
    "\\end{matrix} \\right)\n",
    "$$\n",
    "\n",
    "L'intérêt de cette représentation est que toute transformation devient une multiplication par une matrice. On peut ainsi facilement définir des transformations plus compliquées."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Votre mission, si toutefois vous l'acceptez, consiste des tâches suivantes :\n",
    "\n",
    " 1. Définissez une fonction `point(x, y)` qui rend un tableau avec 3 éléments représentant un point.\n",
    " 2. Définissez une fonction `translation(dx, dy)` qui rend un tableau 3x3 représentant une translation définie par les deux déplacements `dx` et `dy`.\n",
    " 3. Définissez une fonction `rotation(phi)` qui rend un tableau 3x3 représentant une rotation autour de l'origine par un angle `phi`.\n",
    " 4. Utilisez ces fonctions pour construire une rotation par $\\phi=45°$ autour du point $[1, 1]$. La façon la plus simple est d'enchaîner trois transformations :\n",
    "     1. Une translation qui déplace le point $[1, 1]$ à l'origine.\n",
    "     2. Une rotation autour de l'origine.\n",
    "     3. Une translation qui déplace l'origine vers $[1, 1]$.\n",
    "\n",
    "Vérifiez votre code de façon graphique comme montré ci-dessous:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true,
    "hide_input": false
   },
   "outputs": [],
   "source": [
    "def point(x, y):\n",
    "    return np.array([x, y, 1.])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true,
    "hide_input": false
   },
   "outputs": [],
   "source": [
    "def translation(dx, dy):\n",
    "    return np.array([[1., 0., dx],\n",
    "                     [0., 1., dy],\n",
    "                     [0., 0., 1.]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true,
    "hide_input": false
   },
   "outputs": [],
   "source": [
    "def rotation(phi):\n",
    "    return np.array([[np.cos(phi), -np.sin(phi), 0.],\n",
    "                     [np.sin(phi), np.cos(phi),  0.],\n",
    "                     [0.,           0.,          1.]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true,
    "hide_input": false
   },
   "outputs": [],
   "source": [
    "rotation_autour_1_1 = translation(1, 1) @ rotation(45.*np.pi/180.) @ translation(-1, -1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "hide_input": false
   },
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x108cc8be0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(6,6))\n",
    "plt.xlim(0, 4)\n",
    "plt.ylim(0, 4)\n",
    "\n",
    "centre_de_rotation = point(1,1)\n",
    "for p in [point(2, 1), point(1, 2), point(3, 2)]:\n",
    "    p_tr = rotation_autour_1_1 @ p\n",
    "    points_a_connecter = [p, centre_de_rotation, p_tr]\n",
    "    plt.plot([r[0] for r in points_a_connecter],\n",
    "             [r[1] for r in points_a_connecter])"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
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  "language_info": {
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    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.1"
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 },
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 "nbformat_minor": 2
}