{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "toc": "true"
   },
   "source": [
    "# Table of Contents\n",
    " <p><div class=\"lev1 toc-item\"><a href=\"#Oraux-CentraleSupélec-PSI---Juin-2017\" data-toc-modified-id=\"Oraux-CentraleSupélec-PSI---Juin-2017-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>Oraux CentraleSupélec PSI - Juin 2017</a></div><div class=\"lev2 toc-item\"><a href=\"#Remarques-préliminaires\" data-toc-modified-id=\"Remarques-préliminaires-11\"><span class=\"toc-item-num\">1.1&nbsp;&nbsp;</span>Remarques préliminaires</a></div><div class=\"lev2 toc-item\"><a href=\"#Planche-158\" data-toc-modified-id=\"Planche-158-12\"><span class=\"toc-item-num\">1.2&nbsp;&nbsp;</span>Planche 158</a></div><div class=\"lev2 toc-item\"><a href=\"#Planche-162\" data-toc-modified-id=\"Planche-162-13\"><span class=\"toc-item-num\">1.3&nbsp;&nbsp;</span>Planche 162</a></div><div class=\"lev2 toc-item\"><a href=\"#Planche-170\" data-toc-modified-id=\"Planche-170-14\"><span class=\"toc-item-num\">1.4&nbsp;&nbsp;</span>Planche 170</a></div><div class=\"lev1 toc-item\"><a href=\"#À-voir-aussi\" data-toc-modified-id=\"À-voir-aussi-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>À voir aussi</a></div><div class=\"lev2 toc-item\"><a href=\"#Les-oraux---(exercices-de-maths-avec-Python)\" data-toc-modified-id=\"Les-oraux---(exercices-de-maths-avec-Python)-21\"><span class=\"toc-item-num\">2.1&nbsp;&nbsp;</span><a href=\"http://perso.crans.org/besson/infoMP/oraux/solutions/\" target=\"_blank\">Les oraux</a>   <em>(exercices de maths avec Python)</em></a></div><div class=\"lev2 toc-item\"><a href=\"#Fiches-de-révisions-pour-les-oraux\" data-toc-modified-id=\"Fiches-de-révisions-pour-les-oraux-22\"><span class=\"toc-item-num\">2.2&nbsp;&nbsp;</span>Fiches de révisions <em>pour les oraux</em></a></div><div class=\"lev2 toc-item\"><a href=\"#Quelques-exemples-de-sujets-d'oraux-corrigés\" data-toc-modified-id=\"Quelques-exemples-de-sujets-d'oraux-corrigés-23\"><span class=\"toc-item-num\">2.3&nbsp;&nbsp;</span>Quelques exemples de sujets <em>d'oraux</em> corrigés</a></div><div class=\"lev2 toc-item\"><a href=\"#D'autres-notebooks-?\" data-toc-modified-id=\"D'autres-notebooks-?-24\"><span class=\"toc-item-num\">2.4&nbsp;&nbsp;</span>D'autres notebooks ?</a></div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Oraux CentraleSupélec PSI - Juin 2017\n",
    "\n",
    "- Ce [notebook Jupyter](https://www.jupyter.org) est une proposition de correction, en [Python 3](https://www.python.org/), d'exercices d'annales de l'épreuve \"maths-info\" du [concours CentraleSupélec](http://www.concours-centrale-supelec.fr/), filière PSI.\n",
    "- Les exercices viennent de l'[Officiel de la Taupe](http://odlt.fr/), [2016](http://www.odlt.fr/Oraux_2016.pdf) (planches 157 à 173, page 23).\n",
    "- Ce document a été écrit par [Lilian Besson](http://perso.crans.org/besson/), et est disponible en ligne [sur mon site](http://perso.crans.org/besson/infoMP/Oraux_CentraleSupélec_PSI__Juin_2017.html)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Remarques préliminaires\n",
    "- Les exercices sans Python ne sont pas traités.\n",
    "- Les exercices avec Python utilisent Python 3, [numpy](http://numpy.org), [matplotlib](http://matplotlib.org), [scipy](http://scipy.org) et [sympy](http://sympy.org), et essaient d'être résolus le plus simplement et le plus rapidement possible. L'efficacité (algorithmique, en terme de mémoire et de temps de calcul), n'est *pas* une priorité. La concision et simplicité de la solution proposée est prioritaire.\n",
    "- Les modules Python utilisés sont aux [versions suivantes](https://github.com/rasbt/watermark) :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPython 3.5.3\n",
      "IPython 6.1.0\n",
      "\n",
      "scipy 0.19.0\n",
      "numpy 1.12.1\n",
      "matplotlib 2.0.2\n",
      "sympy 1.0\n",
      "seaborn 0.7.1\n",
      "\n",
      "compiler   : GCC 6.3.0 20170118\n",
      "system     : Linux\n",
      "release    : 4.10.0-21-generic\n",
      "machine    : x86_64\n",
      "processor  : x86_64\n",
      "CPU cores  : 4\n",
      "interpreter: 64bit\n",
      "Git hash   : c4118c39f038f15e7dcde2fa4d6a8df0060ae9e4\n"
     ]
    }
   ],
   "source": [
    "%load_ext watermark\n",
    "%watermark -v -m -p scipy,numpy,matplotlib,sympy,seaborn -g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import numpy.linalg as LA\n",
    "import matplotlib as mpl  # inutile\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy as sc        # pas très utile"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pour avoir de belles figures :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import seaborn as sns\n",
    "sns.set(context=\"notebook\", style=\"darkgrid\", palette=\"hls\", font=\"sans-serif\", font_scale=1.4)\n",
    "mpl.rcParams['figure.figsize'] = (19.80, 10.80)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "## Planche 158\n",
    "\n",
    "On donne $f_n(t) = \\frac{1 - \\cos\\left(\\frac{t}{n}\\right)}{t^2(1+t^2)}$.\n",
    "\n",
    "- Tracer avec Python les courbes de $f_n$ pour $n \\in \\{1, \\dots, 10\\}$, ainsi que la fonction constante $y = \\frac{1}{2}$, sur $]0, \\pi[$."
   ]
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6DeUYMkaSVzkLZ8p96IDZkQAAAAAAAYySByhDtqbNFTZwuFRYoJwXpsmdftTs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\nKDVnazy2K5OSQ1mZ5KRkkhFHS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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fcd47ffefd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f(n, t):\n",
    "    return (1 - np.cos(t / n)) / (t**2 *  (1 + t**2))\n",
    "\n",
    "def y(t):\n",
    "    return 0.5 * np.ones_like(t)\n",
    "\n",
    "eps = 1e-5\n",
    "t = np.linspace(0 + eps, np.pi - eps, 1000)\n",
    "plt.figure()\n",
    "for n in range(1, 1 + 10):\n",
    "    plt.plot(t, f(n, t), label=r'$f_{%i}(t)$' % n)\n",
    "plt.plot(t, y(t), label=r'$\\frac{1}{2}$')\n",
    "plt.legend()\n",
    "plt.title(\"Courbe demandée\")\n",
    "plt.xlabel(r\"$]0, \\pi[$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$f_n(t)$ est bien sûr intégrable sur $[1, +\\infty]$, mais c'est moins évident sur $]0, 1]$.\n",
    "La courbe précédente laisse suggérer qu'elle l'est, il faudrait le prouver.\n",
    "\n",
    "(un développement limité montre qu'en $0$, $f_n(t)$ est prolongeable par continuité en fait)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Calculer les $30$ premiers termes de la suite de terme général $u_n = \\int_0^{+\\infty} f_n(t) \\mathrm{d}t$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "- Pour n = 1 \t u_n = 0.5778636758673165\n",
      "- Pour n = 2 \t u_n = 0.1673379686913233\n",
      "- Pour n = 3 \t u_n = 0.07832719999247961\n",
      "- Pour n = 4 \t u_n = 0.04524016647527334\n",
      "- Pour n = 5 \t u_n = 0.0294221986520279\n",
      "- Pour n = 6 \t u_n = 0.02065344571531756\n",
      "- Pour n = 7 \t u_n = 0.015291770261272906\n",
      "- Pour n = 8 \t u_n = 0.011776107184161803\n",
      "- Pour n = 9 \t u_n = 0.009346910316513675\n",
      "- Pour n = 10 \t u_n = 0.007598599569591147\n",
      "- Pour n = 11 \t u_n = 0.006298591120713312\n",
      "- Pour n = 12 \t u_n = 0.005305713621776519\n",
      "- Pour n = 13 \t u_n = 0.004530420616071271\n",
      "- Pour n = 14 \t u_n = 0.003913403332067795\n",
      "- Pour n = 15 \t u_n = 0.0034143635712630925\n",
      "- Pour n = 16 \t u_n = 0.00300503233562181\n",
      "- Pour n = 17 \t u_n = 0.002665127564756733\n",
      "- Pour n = 18 \t u_n = 0.0023797953460884366\n",
      "- Pour n = 19 \t u_n = 0.002137946272152306\n",
      "- Pour n = 20 \t u_n = 0.0019311752431841328\n",
      "- Pour n = 21 \t u_n = 0.0017530128954453457\n",
      "- Pour n = 22 \t u_n = 0.0015984135117823957\n",
      "- Pour n = 23 \t u_n = 0.001463398620721792\n",
      "- Pour n = 24 \t u_n = 0.001344791379603347\n",
      "- Pour n = 25 \t u_n = 0.0012400485222443885\n",
      "- Pour n = 26 \t u_n = 0.0011470784838279937\n",
      "- Pour n = 27 \t u_n = 0.001064184893274427\n",
      "- Pour n = 28 \t u_n = 0.000989962863081455\n",
      "- Pour n = 29 \t u_n = 0.0009232437583422769\n",
      "- Pour n = 30 \t u_n = 0.0008630488183994005\n"
     ]
    }
   ],
   "source": [
    "from scipy.integrate import quad as integral\n",
    "\n",
    "def u_n(n):\n",
    "    def f_n(t):\n",
    "        return f(n, t)\n",
    "    return integral(f_n, 0, np.inf)[0]\n",
    "\n",
    "for n in range(1, 1 + 30):\n",
    "    print(\"- Pour n =\", n, \"\\t u_n =\", u_n(n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Le terme $u_n$ semble tendre vers $0$ pour $n\\to +\\infty$.\n",
    "On le prouverait avec le théorème de convergence dominée (à faire)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Soit $F(x) = \\int_0^{+\\infty} \\frac{1 - \\cos(xt)}{t^2 (1+t^2)} \\mathrm{d}t$.\n",
    "\n",
    "- L'intégrande est évidemment intégrable sur $[1,+\\infty[$ par comparaison (et comme $t\\mapsto \\frac{1}{t^4}$ l'est).\n",
    "- Sur $]0,1]$, $1-\\cos(xt) \\sim_{x\\to 0} \\frac{(xt)^2}{2} $, donc l'intégrande est $\\sim \\frac{x^2}{2(1+t^2)}$ qui est bien intégrable (la constante $\\frac{x^2}{2}$ sort de l'intégrale).\n",
    "Donc $F$ est bien définie sur $R_+^*$.\n",
    "\n",
    "Elle est continue par application directe du théorème de continuité sous le signe intégrale.\n",
    "\n",
    "Elle est prolongeable par continuité en $0$, par $F(0) := 0$ grâce à l'observation précédente : $F(x) \\sim \\frac{x^2}{2} \\int_0^{+\\infty} \\frac{1}{1+t^2} \\mathrm{d}t \\to 0$ quand $x\\to 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fcd4243d6d8>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/local/lib/python3.5/dist-packages/scipy/integrate/quadpack.py:364: IntegrationWarning: The maximum number of subdivisions (50) has been achieved.\n",
      "  If increasing the limit yields no improvement it is advised to analyze \n",
      "  the integrand in order to determine the difficulties.  If the position of a \n",
      "  local difficulty can be determined (singularity, discontinuity) one will \n",
      "  probably gain from splitting up the interval and calling the integrator \n",
      "  on the subranges.  Perhaps a special-purpose integrator should be used.\n",
      "  warnings.warn(msg, IntegrationWarning)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fcd2ccf3a90>]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x7fcd2ccd7470>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Jnjgmu3SuEn/ZJfl8Ct49VtboSTIsy+tpQEYi6gAAAAAAquXaEdmrlyi2cY3k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\nSdTxPE8vvPCCnnrqKf3Xf/1XKp4SAIBv5FWfU3T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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fcd4243d6d8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def F(x):\n",
    "    def f_inf(t):\n",
    "        return (1 - np.cos(x * t)) / (t**2 * (1 + t**2))\n",
    "    return integral(f_inf, 0, np.inf)[0]\n",
    "\n",
    "eps = 1e-4\n",
    "x = np.linspace(0 + eps, 10, 1000)\n",
    "plt.figure()\n",
    "plt.plot(x, np.vectorize(F)(x))\n",
    "plt.title(\"$F(x)$ pour $x = 0 .. 10$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On constate sur la figure que $F$ est bien prolongeable par continuité en $0$.\n",
    "\n",
    "On montrerait aussi que $F$ est de classe $\\mathcal{C}^1$ facilement, par application directe du théorème de dérivation généralisée sous le signe intégral."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "## Planche 162\n",
    "\n",
    "Soit $(P_n)_{n\\geq 0}$ une suite de polynômes définis par $P_0 = 1$, $P_1 = 2X$ et $P_{n+1} = 2 X P_n - P_{n-1}$. Calculons $P_2,\\dots,P_8$.\n",
    "\n",
    "On pourrait tout faire avec des listes gérées manuellement, mais c'est assez compliqué.\n",
    "\n",
    "Il vaut mieux aller vite, en utilisant le module [numpy.polynomial](https://docs.scipy.org/doc/numpy/reference/routines.polynomials.classes.html)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "code_folding": [
     0,
     35
    ],
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Ce morceau est juste là pour avoir un joli rendu\n",
    "def Polynomial_to_LaTeX(p):\n",
    "    \"\"\" Small function to print nicely the polynomial p as we write it in maths, in LaTeX code.\n",
    "    \n",
    "    - Source: https://nbviewer.jupyter.org/github/Naereen/notebooks/blob/master/Demonstration%20of%20numpy.polynomial.Polynomial%20and%20nice%20display%20with%20LaTeX%20and%20MathJax%20%28python3%29.ipynb\n",
    "    \"\"\"\n",
    "    coefs = p.coef  # List of coefficient, sorted by increasing degrees\n",
    "    res = \"\"  # The resulting string\n",
    "    for i, a in enumerate(coefs):\n",
    "        if int(a) == a:  # Remove the trailing .0\n",
    "            a = int(a)\n",
    "        if i == 0:  # First coefficient, no need for X\n",
    "            if a > 0:\n",
    "                res += \"{a} + \".format(a=a)\n",
    "            elif a < 0:  # Negative a is printed like (a)\n",
    "                res += \"({a}) + \".format(a=a)\n",
    "            # a = 0 is not displayed \n",
    "        elif i == 1:  # Second coefficient, only X and not X**i\n",
    "            if a == 1:  # a = 1 does not need to be displayed\n",
    "                res += \"X + \"\n",
    "            elif a > 0:\n",
    "                res += \"{a} \\;X + \".format(a=a)\n",
    "            elif a < 0:\n",
    "                res += \"({a}) \\;X + \".format(a=a)\n",
    "        else:\n",
    "            if a == 1:\n",
    "                # A special care needs to be addressed to put the exponent in {..} in LaTeX\n",
    "                res += \"X^{i} + \".format(i=\"{%d}\" % i)\n",
    "            elif a > 0:\n",
    "                res += \"{a} \\;X^{i} + \".format(a=a, i=\"{%d}\" % i)\n",
    "            elif a < 0:\n",
    "                res += \"({a}) \\;X^{i} + \".format(a=a, i=\"{%d}\" % i)\n",
    "    if res == \"\":\n",
    "        res = \"0000\"\n",
    "    return \"$\" + res[:-3] + \"$\"\n",
    "\n",
    "def setup_prrint():\n",
    "    ip = get_ipython()\n",
    "    latex_formatter = ip.display_formatter.formatters['text/latex']\n",
    "    latex_formatter.for_type_by_name('numpy.polynomial.polynomial',\n",
    "                                     'Polynomial', Polynomial_to_LaTeX)\n",
    "\n",
    "setup_prrint()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Je recommande d'importer `numpy.polynomial.Polynomial` et de l'appeller `P`.\n",
    "Définir directement le monôme $X$ comme `P([0, 1])`, donné par la liste de ses coefficients $[a_k]_{0 \\leq k \\leq \\delta(X)} = [0, 1]$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$X$"
      ],
      "text/plain": [
       "Polynomial([ 0.,  1.], [-1,  1], [-1,  1])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from numpy.polynomial import Polynomial as P\n",
    "X = P([0, 1])\n",
    "X"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ensuite, on peut rapidement écrire une fonction, qui donne $P_n$ pour un $n \\geq 0$.\n",
    "Pas besoin d'être malin, on recalcule tout dans la fonction.\n",
    "\n",
    "- `Pnm1` signifie $P_{n - 1}$\n",
    "- `Pnext` signifie $P_{n + 1}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pour n = 0 P_n =\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$1$"
      ],
      "text/plain": [
       "Polynomial([ 1.], [-1,  1], [-1,  1])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pour n = 1 P_n =\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$2 \\;X$"
      ],
      "text/plain": [
       "Polynomial([ 0.,  2.], [-1,  1], [-1,  1])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pour n = 2 P_n =\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$(-1) + 4 \\;X^{2}$"
      ],
      "text/plain": [
       "Polynomial([-1.,  0.,  4.], [-1.,  1.], [-1.,  1.])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pour n = 3 P_n =\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$(-4) \\;X + 8 \\;X^{3}$"
      ],
      "text/plain": [
       "Polynomial([ 0., -4.,  0.,  8.], [-1.,  1.], [-1.,  1.])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pour n = 4 P_n =\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$1 + (-12) \\;X^{2} + 16 \\;X^{4}$"
      ],
      "text/plain": [
       "Polynomial([  1.,   0., -12.,   0.,  16.], [-1.,  1.], [-1.,  1.])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pour n = 5 P_n =\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$6 \\;X + (-32) \\;X^{3} + 32 \\;X^{5}$"
      ],
      "text/plain": [
       "Polynomial([  0.,   6.,   0., -32.,   0.,  32.], [-1.,  1.], [-1.,  1.])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pour n = 6 P_n =\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$(-1) + 24 \\;X^{2} + (-80) \\;X^{4} + 64 \\;X^{6}$"
      ],
      "text/plain": [
       "Polynomial([ -1.,   0.,  24.,   0., -80.,   0.,  64.], [-1.,  1.], [-1.,  1.])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pour n = 7 P_n =\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$(-8) \\;X + 80 \\;X^{3} + (-192) \\;X^{5} + 128 \\;X^{7}$"
      ],
      "text/plain": [
       "Polynomial([   0.,   -8.,    0.,   80.,    0., -192.,    0.,  128.], [-1.,  1.], [-1.,  1.])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pour n = 8 P_n =\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$1 + (-40) \\;X^{2} + 240 \\;X^{4} + (-448) \\;X^{6} + 256 \\;X^{8}$"
      ],
      "text/plain": [
       "Polynomial([   1.,    0.,  -40.,    0.,  240.,    0., -448.,    0.,  256.], [-1.,  1.], [-1.,  1.])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def P_n(n):\n",
    "    P0 = P([1])\n",
    "    P1 = P([0, 2])\n",
    "    Pnm1, Pn = P0, P1\n",
    "    for i in range(n):\n",
    "        Pnext = (2 * X * Pn) - Pnm1\n",
    "        Pnm1, Pn = Pn, Pnext\n",
    "    return Pnm1\n",
    "\n",
    "for n in range(0, 1 + 8):\n",
    "    print(\"Pour n =\", n, \"P_n =\")\n",
    "    P_n(n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Premières observations :\n",
    "- Le dégré de $P_n$ est $n$,\n",
    "- Son coefficient dominant est $2^{n-1}$ si $n>0$,\n",
    "- Sa parité est impaire si $n$ est pair, paire si $n$ est impair.\n",
    "\n",
    "Ces trois points se montrent assez rapidement par récurrence simple, à partir de $P_0,P_1$ et la relation de récurrence définissant $P_n$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On vérifie mathématiquement que $\\langle P, Q \\rangle := \\frac{2}{\\pi} \\int_{-1}^{1} \\sqrt{1-t^2} P(t) Q(t) \\mathrm{d}t$ est un produit scalaire pour les polynômes réels.\n",
    "(il est évidemment bien défini puisque la racine carrée existe, et que les fonctions intégrées sont de continues sur $[-1,1]$, symétrique, positif si $P=Q$, et il est défini parce que $P^2(t) \\geq 0$).\n",
    "\n",
    "Calculons $\\langle P_i, P_j \\rangle$ pour $0 \\leq i,j \\leq 8$.\n",
    "L'intégration est faite *numériquement*, avec [`scipy.integrate.quad`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.quad.html)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.integrate import quad\n",
    "\n",
    "def produit_scalaire(P, Q):\n",
    "    def f(t):\n",
    "        return np.sqrt(1 - t**2) * P(t) * Q(t)\n",
    "    return (2 / np.pi) * quad(f, -1, 1)[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "< P_1, P_1 > = 1\n",
      "< P_1, P_2 > = 0\n",
      "< P_1, P_3 > = -0\n",
      "< P_1, P_4 > = 0\n",
      "< P_1, P_5 > = -0\n",
      "< P_1, P_6 > = 0\n",
      "< P_1, P_7 > = -0\n",
      "< P_1, P_8 > = 0\n",
      "< P_2, P_2 > = 1\n",
      "< P_2, P_3 > = 0\n",
      "< P_2, P_4 > = -0\n",
      "< P_2, P_5 > = 0\n",
      "< P_2, P_6 > = -0\n",
      "< P_2, P_7 > = 0\n",
      "< P_2, P_8 > = -0\n",
      "< P_3, P_3 > = 1\n",
      "< P_3, P_4 > = 0\n",
      "< P_3, P_5 > = -0\n",
      "< P_3, P_6 > = 0\n",
      "< P_3, P_7 > = -0\n",
      "< P_3, P_8 > = 0\n",
      "< P_4, P_4 > = 1\n",
      "< P_4, P_5 > = 0\n",
      "< P_4, P_6 > = -0\n",
      "< P_4, P_7 > = 0\n",
      "< P_4, P_8 > = -0\n",
      "< P_5, P_5 > = 1\n",
      "< P_5, P_6 > = 0\n",
      "< P_5, P_7 > = -0\n",
      "< P_5, P_8 > = 0\n",
      "< P_6, P_6 > = 1\n",
      "< P_6, P_7 > = 0\n",
      "< P_6, P_8 > = -0\n",
      "< P_7, P_7 > = 1\n",
      "< P_7, P_8 > = 0\n",
      "< P_8, P_8 > = 1\n"
     ]
    }
   ],
   "source": [
    "# on calcule qu'une seule fois\n",
    "P_n_s = [P_n(n) for n in range(0, 1 + 8)]\n",
    "\n",
    "for i in range(1, 1 + 8):\n",
    "    for j in range(i, 1 + 8):\n",
    "        Pi, Pj = P_n_s[i], P_n_s[j]\n",
    "        ps = np.round(produit_scalaire(Pi, Pj), 8)\n",
    "        print(\"< P_{}, P_{} > = {:.3g}\".format(i, j, ps))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 0, 0, 0, 0, 0, 0, 0],\n",
       "       [0, 1, 0, 0, 0, 0, 0, 0],\n",
       "       [0, 0, 1, 0, 0, 0, 0, 0],\n",
       "       [0, 0, 0, 1, 0, 0, 0, 0],\n",
       "       [0, 0, 0, 0, 1, 0, 0, 0],\n",
       "       [0, 0, 0, 0, 0, 1, 0, 0],\n",
       "       [0, 0, 0, 0, 0, 0, 1, 0],\n",
       "       [0, 0, 0, 0, 0, 0, 0, 1]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "produits_scalaires = np.zeros((8, 8))\n",
    "\n",
    "for i in range(1, 1 + 8):\n",
    "    for j in range(i, 1 + 8):\n",
    "        Pi, Pj = P_n_s[i], P_n_s[j]\n",
    "        produits_scalaires[i - 1, j - 1] = np.round(produit_scalaire(Pi, Pj), 8)\n",
    "\n",
    "produits_scalaires.astype(int)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La famille $(P_i)_{0 \\leq i \\leq 8}\\;$  est *orthogonale*.\n",
    "(les `-0` sont des `0`, la différence vient des erreurs d'arrondis)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Soit $\\Phi(P) = 3XP' - (1-X^2)P''$ (erreur dans l'énoncé, le deuxième terme est évidemment $P''$ et non $P'$).\n",
    "Elle conserve (ou diminue) le degré de $P$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def Phi(P):\n",
    "    return 3 * X * P.deriv() - (1 - X**2) * P.deriv(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On calcule sa matrice de passage, dans la base $(P_i)_{1\\leq i \\leq 8}$ :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# on calcule qu'une seule fois\n",
    "P_n_s = [P_n(n) for n in range(0, 1 + 8)]\n",
    "\n",
    "matrice_Phi = [\n",
    "    [\n",
    "        np.round(produit_scalaire(Phi(P_n_s[i]), P_n_s[j]), 8)\n",
    "        for i in range(1, 1 + 8)\n",
    "    ] for j in range(1, 1 + 8)\n",
    "]\n",
    "matrice_Phi = np.array(matrice_Phi, dtype=int)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(8, 8)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "array([[ 3,  0,  0,  0,  0,  0,  0,  0],\n",
       "       [ 0,  8,  0,  0,  0,  0,  0,  0],\n",
       "       [ 0,  0, 15,  0,  0,  0,  0,  0],\n",
       "       [ 0,  0,  0, 24,  0,  0,  0,  0],\n",
       "       [ 0,  0,  0,  0, 35,  0,  0,  0],\n",
       "       [ 0,  0,  0,  0,  0, 48,  0,  0],\n",
       "       [ 0,  0,  0,  0,  0,  0, 63,  0],\n",
       "       [ 0,  0,  0,  0,  0,  0,  0, 80]])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrice_Phi.shape\n",
    "matrice_Phi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Elle est diagonale ! Et trivialement inversible !"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "73156608000.0"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.linalg import det\n",
    "det(matrice_Phi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cette matrice est inversible, donc dans la base $(P_i)_{1\\leq i \\leq 8}$, l'application linéaire $\\Phi$ est une bijection.\n",
    "\n",
    "On peut même dire plus : en renormalisant les $P_i$, on peut faire de $\\Phi$ l'identité..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 0, 0, 0, 0, 0, 0, 0],\n",
       "       [0, 1, 0, 0, 0, 0, 0, 0],\n",
       "       [0, 0, 1, 0, 0, 0, 0, 0],\n",
       "       [0, 0, 0, 1, 0, 0, 0, 0],\n",
       "       [0, 0, 0, 0, 1, 0, 0, 0],\n",
       "       [0, 0, 0, 0, 0, 1, 0, 0],\n",
       "       [0, 0, 0, 0, 0, 0, 1, 0],\n",
       "       [0, 0, 0, 0, 0, 0, 0, 1]])"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P_n_s_normalises = np.asarray(P_n_s[1:]) / np.sqrt(matrice_Phi.diagonal())\n",
    "\n",
    "matrice_Phi_normalise = [\n",
    "    [\n",
    "        np.round(produit_scalaire(Phi(P_n_s_normalises[i - 1]), P_n_s_normalises[j - 1]), 8)\n",
    "        for i in range(1, 1 + 8)\n",
    "    ] for j in range(1, 1 + 8)\n",
    "]\n",
    "matrice_Phi_normalise = np.array(matrice_Phi_normalise)\n",
    "\n",
    "matrice_Phi_normalise.astype(int)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On peut utiliser ce fait pour montrer, par deux intégrations par parties, le résultat annoncé sur l'orthogonalité de la famille $(P_i)_{1\\leq i \\leq 8}\\;$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "## Planche 170\n",
    "\n",
    "On étudie le comportement d'une particule évoluant sur 4 états, avec certaines probabilités :\n",
    "\n",
    "![centrale2017002_planche170.png](centrale2017002_planche170.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On fixe la constante $p = \\frac12$ dès maintenant, on définit la matrice de transition $A$, telle que définie un peu après dans l'exercice.\n",
    "\n",
    "Les états sont représentés par `[0, 1, 2, 3]` plutôt que $A_0, A_1, A_2, A_3$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "p = 0.5\n",
    "\n",
    "A = np.array([\n",
    "    [1, 0, 0, 0],\n",
    "    [p, 0, 1-p, 0],\n",
    "    [0, p, 0, 1-p],\n",
    "    [0, 0, 0, 1]\n",
    "])\n",
    "etats = [0, 1, 2, 3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy.random as rd"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Une transition se fait en choisissant un état $x_{n+1}$ parmi $\\{0, 1, 2, 3\\}$, avec probabilité $\\mathbb{P}(x_{n+1} = k) = A_{x_n, k}$.\n",
    "La fonction [`numpy.random.choice`](https://docs.scipy.org/doc/numpy/reference/generated/numpy.random.choice.html) fait ça directement."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def une_transition(xn):\n",
    "    return rd.choice(etats, p = A[xn])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "une_transition(0)\n",
    "une_transition(1)\n",
    "une_transition(2)\n",
    "une_transition(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On peut écrire la fonction à la main, comme :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def une_transition_longue(xn):\n",
    "    if xn == 0 or xn == 3:\n",
    "        return xn\n",
    "    elif xn == 1:\n",
    "        if rd.random() < p:\n",
    "            return 0  # avec probabilité p\n",
    "        else:\n",
    "            return 2  # avec probabilité 1-p\n",
    "    elif xn == 2:\n",
    "        if rd.random() < p:\n",
    "            return 1\n",
    "        else:\n",
    "            return 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "une_transition_longue(0)\n",
    "une_transition_longue(1)\n",
    "une_transition_longue(2)\n",
    "une_transition_longue(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Faire plusieurs transitions se fait juste en appliquant la même fonction $n$ fois."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def n_transitions(n, x0):\n",
    "    x = x0\n",
    "    for i in range(n):\n",
    "        x = une_transition(x)\n",
    "    return x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_transitions(10, 0)\n",
    "n_transitions(10, 1)\n",
    "n_transitions(10, 2)\n",
    "n_transitions(10, 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Faisons $N=1000$ répétitions de cette expérience, à l'horizon disons $n=100$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "n = 100\n",
    "N = 1000\n",
    "\n",
    "def histogramme(n, N, x0):\n",
    "    observations = np.zeros(len(etats))\n",
    "    for experience in range(N):\n",
    "        obs = n_transitions(n, x0)\n",
    "        observations[obs] += 1\n",
    "    plt.bar(etats, observations)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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CoVAYPHhw4T//8z9bjl23bl1h4sSJhaFDhxa+9KX/b+eOURMKoiiA\nks1YyHMBlvJBhFRWrsPFuA7XYP1/4SJ+4Sq8aSNpDCQIvnNg+ikuj5kLM5+5XC6v2jb/5Nk83O/3\nnE6nbDabVFV2u91DVngP2+02VZXlcpnFYpGqSlVlnmfzoaFn82A+9DCO40MOvq95nn+cIadpyn6/\nT1VlGIacz+cX7p6/9ts8uFO8v+v1msPhkNVqlfV6nePxmNvtlsQdo6Nn8/CKM8RH4lcrAAAAgG5a\nPx8DAAAA6EopBAAAANCQUggAAACgIaUQAAAAQENKIQAAAICGlEIAAAAADSmFAAAAABpSCgEAAAA0\npBQCAAAAaOgLDZzif4QbC2IAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fcd2b8b4cf8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fcd2ccffef0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fcd423e8e80>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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bzAPFzAPFzAP7MxMUMw8U64h52NXuZ4A8OvIa3llfL9o0CtXW1sasWbPimmuu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\nUggAAACgoG9HduJ/E2754QAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fcd2cca0198>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "histogramme(n, N, 0)\n",
    "histogramme(n, N, 1)\n",
    "histogramme(n, N, 2)\n",
    "histogramme(n, N, 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Mathématiquement, sur papier on calcule le polynôme caractéristique de $A$, et on vérifie qu'il est scindé ssi $p \\neq 0, 1$ (mais pas à racine simple)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pour diagonaliser, on utile le module `numpy.linalg`, et la fonction [`numpy.linalg.eig`](https://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.eig.html)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from numpy import linalg as LA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1. ,  0.5,  0. ,  0. ],\n",
       "       [ 0. ,  0. ,  0.5,  0. ],\n",
       "       [ 0. ,  0.5,  0. ,  0. ],\n",
       "       [ 0. ,  0. ,  0.5,  1. ]])"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = A.T\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1. ,  1. ,  0.5, -0.5])"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "array([[ 1.        ,  0.        , -0.5       , -0.2236068 ],\n",
       "       [ 0.        ,  0.        ,  0.5       ,  0.67082039],\n",
       "       [ 0.        ,  0.        ,  0.5       , -0.67082039],\n",
       "       [ 0.        ,  1.        , -0.5       ,  0.2236068 ]])"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spectre, matricePassage = LA.eig(A)\n",
    "spectre\n",
    "matricePassage"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ici, on vérifie que le spectre contient deux fois la valeur propre $1$, qui vient des deux puits $A_0,A_3$, et deux valeurs symétriques.\n",
    "\n",
    "On peut vérifier que $A = P \\Lambda P^{-1}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  1.00000000e+00,   5.00000000e-01,  -6.14248029e-17,\n",
       "          0.00000000e+00],\n",
       "       [  0.00000000e+00,  -4.08433854e-17,   5.00000000e-01,\n",
       "          0.00000000e+00],\n",
       "       [  0.00000000e+00,   5.00000000e-01,   1.34656917e-16,\n",
       "          0.00000000e+00],\n",
       "       [  0.00000000e+00,   5.78726134e-17,   5.00000000e-01,\n",
       "          1.00000000e+00]])"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "array([[  1.00000000e+00,   5.00000000e-01,  -9.74554924e-17,\n",
       "          0.00000000e+00],\n",
       "       [  0.00000000e+00,  -4.49809423e-17,   5.00000000e-01,\n",
       "          0.00000000e+00],\n",
       "       [  0.00000000e+00,   5.00000000e-01,   1.30519360e-16,\n",
       "          0.00000000e+00],\n",
       "       [  0.00000000e+00,   4.95974995e-17,   5.00000000e-01,\n",
       "          1.00000000e+00]])"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Lambda = np.diag(spectre)\n",
    "matricePassageinv = LA.inv(matricePassage)\n",
    "\n",
    "# avec Python >= 3.6\n",
    "matricePassage @ Lambda @ matricePassageinv\n",
    "# avant 3.6\n",
    "matricePassage.dot(Lambda.dot(matricePassageinv))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sans erreur d'arrondis, ça donne :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1. ,  0.5, -0. ,  0. ],\n",
       "       [ 0. , -0. ,  0.5,  0. ],\n",
       "       [ 0. ,  0.5,  0. ,  0. ],\n",
       "       [ 0. ,  0. ,  0.5,  1. ]])"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "array([[ 1. ,  0.5, -0. ,  0. ],\n",
       "       [ 0. , -0. ,  0.5,  0. ],\n",
       "       [ 0. ,  0.5,  0. ,  0. ],\n",
       "       [ 0. ,  0. ,  0.5,  1. ]])"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.round(matricePassage @ Lambda @ matricePassageinv, 3)\n",
    "np.round(matricePassage.dot(Lambda.dot(matricePassageinv)), 3)\n",
    "\n",
    "np.all(np.round(matricePassage @ Lambda @ matricePassageinv, 3) == A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On peut ensuite calculer $\\lim_{n\\to\\infty} X_n$ en calculant\n",
    "$P \\Lambda' P^{-1} X_0$ si $\\Lambda' := \\lim_{n\\to\\infty} \\Lambda^n = \\mathrm{Diag}(\\lim_{n\\to\\infty} \\lambda_i^n)$ qui existe bien puisque $\\mathrm{Sp}(A) = \\{1, \\pm\\sqrt{p(1-p)}\\} \\subset [-1,1]$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 0, 0, 0],\n",
       "       [0, 1, 0, 0],\n",
       "       [0, 0, 0, 0],\n",
       "       [0, 0, 0, 0]])"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def limite_inf(t):\n",
    "    if t <= -1:\n",
    "        raise ValueError(\"Pas de limite\")\n",
    "    elif -1 < t < 1:\n",
    "        return 0\n",
    "    elif t == 1:\n",
    "        return 1\n",
    "    else:\n",
    "        return np.inf\n",
    "\n",
    "LambdaInf = np.diag([limite_inf(lmbda) for lmbda in spectre])\n",
    "LambdaInf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pour X0 = [ 1.  0.  0.  0.]\n",
      "    => limite Xn pour n -> oo = [ 1.  0.  0.  0.]\n",
      "Pour X0 = [ 0.  1.  0.  0.]\n",
      "    => limite Xn pour n -> oo = [ 0.66666667  0.          0.          0.33333333]\n",
      "Pour X0 = [ 0.  0.  1.  0.]\n",
      "    => limite Xn pour n -> oo = [ 0.33333333  0.          0.          0.66666667]\n",
      "Pour X0 = [ 0.  0.  0.  1.]\n",
      "    => limite Xn pour n -> oo = [ 0.  0.  0.  1.]\n"
     ]
    }
   ],
   "source": [
    "for x0 in etats:\n",
    "    X0 = np.zeros(len(etats))\n",
    "    X0[x0] = 1\n",
    "    print(\"Pour X0 =\", X0)\n",
    "    Xinf = (matricePassage @ LambdaInf @ matricePassageinv) @ X0\n",
    "    print(\"    => limite Xn pour n -> oo =\", Xinf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ça correspond exactement aux histogrammes obtenus plus haut.\n",
    "Peu importe l'état initial, la particule finira dans un des deux puits.\n",
    "(C'est ce qu'on appelle des états absorbants)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "# À voir aussi\n",
    "\n",
    "## [Les oraux](http://perso.crans.org/besson/infoMP/oraux/solutions/)   *(exercices de maths avec Python)*\n",
    "\n",
    "Se préparer aux oraux de [\"maths avec Python\" (maths 2)](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/#oMat2) du concours Centrale Supélec peut être utile.\n",
    "\n",
    "Après les écrits et la fin de l'année, pour ceux qui seront admissibles à Centrale-Supélec, ils vous restera <b>les oraux</b> (le concours Centrale-Supélec a un <a title=\"Quelques exemples d'exercices sur le site du concours Centrale-Supélec\" href=\"http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/#oMat2\">oral d'informatique</a>, et un peu d'algorithmique et de Python peuvent en théorie être demandés à chaque oral de maths et de SI).\n",
    "\n",
    "Je vous invite à lire [cette page avec attention](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/#oMat2), et à jeter un œil aux documents mis à disposition :\n",
    "\n",
    "## Fiches de révisions *pour les oraux*\n",
    "\n",
    "1. [Calcul matriciel](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/Python-matrices.pdf), avec [numpy](https://docs.scipy.org/doc/numpy/) et [numpy.linalg](http://docs.scipy.org/doc/numpy/reference/routines.linalg.html),\n",
    "2. [Réalisation de tracés](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/Python-plot.pdf), avec [matplotlib](http://matplotlib.org/users/beginner.html),\n",
    "3. [Analyse numérique](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/Python-AN.pdf), avec [numpy](https://docs.scipy.org/doc/numpy/) et [scipy](http://docs.scipy.org/doc/scipy/reference/tutorial/index.html). Voir par exemple [scipy.integrate](http://docs.scipy.org/doc/scipy/reference/tutorial/integrate.html) avec les fonctions [scipy.integrate.quad](http://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.quad.html) (intégrale numérique) et [scipy.integrate.odeint](http://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.odeint.html) (résolution numérique d'une équation différentielle),\n",
    "4. [Polynômes](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/Python-polynomes.pdf) : avec [numpy.polynomials](https://docs.scipy.org/doc/numpy/reference/routines.polynomials.package.html), [ce tutoriel peut aider](https://docs.scipy.org/doc/numpy/reference/routines.polynomials.classes.html),\n",
    "5. [Probabilités](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/Python-random.pdf), avec [numpy](https://docs.scipy.org/doc/numpy/) et [random](https://docs.python.org/3/library/random.html).\n",
    "\n",
    "Pour réviser : voir [ce tutoriel Matplotlib (en anglais)](http://www.labri.fr/perso/nrougier/teaching/matplotlib/), [ce tutoriel Numpy (en anglais)](http://www.labri.fr/perso/nrougier/teaching/numpy/numpy.html).\n",
    "Ainsi que tous les [TP](http://perso.crans.org/besson/infoMP/TPs/solutions/), [TD](http://perso.crans.org/besson/infoMP/TDs/solutions/) et [DS](http://perso.crans.org/besson/infoMP/DSs/solutions/) en Python que j'ai donné et corrigé au Lycée Lakanal (Sceaux, 92) en 2015-2016 !\n",
    "\n",
    "## Quelques exemples de sujets *d'oraux* corrigés\n",
    "> Ces 5 sujets sont corrigés, et nous les avons tous traité en classe durant les deux TP de révisions pour les oraux (10 et 11 juin).\n",
    "\n",
    "- PC : [sujet #1](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/PC-Mat2-2015-27.pdf) ([correction PC #1](http://perso.crans.org/besson/infoMP/oraux/solutions/PC_Mat2_2015_27.html)), [sujet #2](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/PC-Mat2-2015-28.pdf) ([correction PC #2](http://perso.crans.org/besson/infoMP/oraux/solutions/PC_Mat2_2015_28.html)).\n",
    "- PSI : [sujet #1](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/PSI-Mat2-2015-24.pdf) ([correction PSI #1](http://perso.crans.org/besson/infoMP/oraux/solutions/PSI_Mat2_2015_24.html)), [sujet #2](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/PSI-Mat2-2015-25.pdf) ([correction PSI #2](http://perso.crans.org/besson/infoMP/oraux/solutions/PSI_Mat2_2015_25.html)), [sujet #3](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/PSI-Mat2-2015-26.pdf) ([correction PSI #3](http://perso.crans.org/besson/infoMP/oraux/solutions/PSI_Mat2_2015_26.html)).\n",
    "- MP : pas de sujet mis à disposition, mais le programme est le même que pour les PC et PSI (pour cette épreuve)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "## D'autres notebooks ?\n",
    "\n",
    "> Ce document est distribué [sous licence libre (MIT)](https://lbesson.mit-license.org/), comme [les autres notebooks](https://GitHub.com/Naereen/notebooks/) que j'ai écrit depuis 2015."
   ]
  }
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