Compte rendu TP1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#                            <font color='orange'>TP1:Analyse_Numérique</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Réalisé par:Ahlem Brahmi 2DNI2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <u>Exercice 2:<u>(methode de dechotomie)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On considère l'équation :$$ f(x)=x^3+x^2-3x-3=0$$1) Dessiner la courbe de $f$ dans l'intervalle $[-2;2]$, puis trouver des intervalles convenables pour appliquer la méthode de >bisection.\n",
    "\n",
    "2) Utiliser un programme dicho.m qui permettra de trouver les solutions de l'équation $f(x)=0$ avec $eps=0.001$.\n",
    "\n",
    "3) Modifier le programme dicho.m pour qu'il donne le nombre d'itération nécessaire pour avoir la solution à $eps$ près."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color='green'>corrigé:</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x19f8771a2b0>]"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from numpy import linspace\n",
    "f=lambda x: x**3+x**2-3*x-3\n",
    "T=linspace(-2,2,41)\n",
    "plt.title(\"solution dans [-2:2] \");\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel (\"f(x)\")\n",
    "plt.grid(True)\n",
    "plt.plot(T,f(T))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## D'aprés le graphe de **f** il existe 3 solutions :\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "   $\\alpha_1 \\in ]-2;-1.5[$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " $\\alpha_2 \\in ]-1.5;-0.5[$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " $\\alpha_3 \\in ]1.5;2[$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Dicho(a,b,f,e):\n",
    " m=(a+b)/2\n",
    "   \n",
    " ecart = abs(b-a)\n",
    " while ecart>e:\n",
    "        if f(a)*f(m)<0:\n",
    "             b=m\n",
    "        else:\n",
    "            a=m\n",
    "        m=(a+b)/2\n",
    "        ecart = b-a\n",
    " return m\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-1.732025146484375\n"
     ]
    }
   ],
   "source": [
    "f=lambda x: x**3+x**2-3*x-3\n",
    "print(Dicho(-2,-1.5,f,0.0001))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Dicho(a,b,f,eps):\n",
    "    m=(a+b)/2\n",
    "    ecart = abs(b-a)\n",
    "    n=0\n",
    "    while abs(b-a)>eps:\n",
    "        n+=1\n",
    "        if f(a)*f(m)<0:\n",
    "             b=m\n",
    "        else:\n",
    "            a=m\n",
    "        m=(a+b)/2\n",
    "    print (\" nbre d'itérations nécessaires :\", n )\n",
    "    return m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " nbre d'itérations nécessaires : 9\n",
      " nbre d'itérations nécessaires : 10\n",
      " nbre d'itérations nécessaires : 9\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "1.73193359375"
      ]
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Dicho(-2,-1.5,f,0.001)\n",
    "Dicho(-1.5,-0.5,f,0.001)\n",
    "Dicho(1.5,2,f,0.001)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color='red'>Conclusion:</font> "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "la méthode de dichotomie permet de connaitre le\n",
    "nombre d’itérations nécessaires en fonction de la précision voulue. Elle est de plus très simple à\n",
    "programmer. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Mais son inconvénient majeur réside dans l’hypothèse du départ f(a)f(b) < 0 qui laisse de\n",
    "coté toutes les fonctions qui s’annulent sans changer de signe."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <u>Exercice 3<u>:(methode de point fixe)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "n considère l'équation :$$ f(x)=x^3+4x^2-10=0$$1) Dessiner la courbe de $f$ dans l'intervalle $[1;2]$, puis trouver une valeur approché de la solution à 0.001 près (En >utilisant la fonction zoom du graphique).\n",
    "\n",
    "2) Vérifier que cette équation est équivalente aux 3 schéma du point fixe suivant:\n",
    "\n",
    "a) $g_1(x)=\\frac{1}{2}\\sqrt{10-x^3}=x$.\n",
    "\n",
    "b) $g_2(x)=\\sqrt{\\displaystyle\\frac{10}{4+x}}=x$.\n",
    "\n",
    "c) $g_3(x)=x-x^3-4x^2+10=x$.\n",
    "\n",
    "3) Ecrire puis appliquer une fonction Matlab que l'on appellera pointfixe.m aux trois schémas précédents en prenant >x0=1.5;eps=0.001.\n",
    "\n",
    "4) Quelle est la fonction (parmi $g_1, g_2$ et $g_3$) qui donne la convergence la plus rapide?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color='green'>Corrigé:</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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aAgtVdZGqbgb6AC0D1hQcquZpctRRluXzuONsuP/aa/4ldMLHkiVw4YVwzjlQrpzli3rzTe+raUzgIw2gMvBLntdLgEb5D4qmjjKEp1ZvSXSUW7CAw7p1Y5/p09lQtSo/PPMMq44/3hK5xXBNYWkTCI+WsOhIWXJyLC7o4Yft+TPPQPv2ULp00MqcBBMGo1FQBJputyGKOsoQnlq9O6Tj55/ty/f++3aH9tpr7H7jjRy1yy7J15JgwqIlLDpSkT3mzTMDMW2alV19/XU49NCgZTlJIgxGYwmQN9dFFWBpQFqSy+rV5jrbubO9vvdei79wH3YnjPz5Jzz8MPW6doWKFc39+7//9cwDGUYYjMZkoIaIHAL8ClwMXBqspASzYYOtUXTsaDmjrrzSaiB7zWMnjKjCRx/BXXfBihX82qoVVd55x+tcZCiBGw1VzRGR24ARQCmgp6pmBywrMWzZAj17moFYuhRatDDDcdRRQStznILJDR794gtzyhg6lIVr11LFDUbGEgbvKVR1mKrWVNXDVPXpoPXEna1b4YMPLDr2ppugWjX48ktzU3SD4YSRjRvhkUfg6KMtLqhrV/jmG4vBcDKaUBiNtGXbNujfH+rWtTKW5cqZO+348XDqqUGrcxKIiLwgIvNEZJaIDBSRvYPWFDVDh1pdi6eegosustHGzTd7qhoHcKORGFQpP3481KtnydpycmzRcNo082f3hcNMYBRQR1WPBr4HHghYT/H89BO0amV9tEwZGDsWeveGChWCVuaECDca8UTV6gPUr0+dRx6xtAq9e0N2Nvzf/3mRmQxCVUeqak7k5UTMKzCcbN4Mzz5r5YBHjrTnM2aAuyQ7BRD4QnhasG0bDBhgw/mZM6F6debedx+1nnoKdvYmdrgW+LiwndEEriYqGHHvadOo0bkzZX/+mRUnn8zCW2/l74oVYcKEQs8JU2BkWLSERQckXov/osVC7rTT009btbyaNaFXL7j0Un4fP55abjDSGhEZDVQsYNdDqjo4csxDQA7wQWH/J5rA1bgHIy5dagF6ffpYYN7QoezfogX7R3FqmAIjw6IlLDog8Vr8V60kbNoE774LL7xg2Tzr1IEPP7QpKF8szBhUtUlR+0XkKuAc4ExV3S7LQSBs2QKvvmqJMLdsgQ4d4L77YLfdglbmpAhuNHaENWssNfkrr1iN44YN4aWX4Nxzfb3C+Rci0gy4DzhNVTcErQew/GW33WZrbC1amPE47LCgVTkphhuNaFiyxFJ99OhhacqbNLGRxemnuyeUUxhdgF2BUWJ9ZKKq3hSIkqVLrc7Fhx9ajNCgQVZ+1fuuUwLcaBTFzJnQqZOlUNi2zaaf7rnHXGkdpwhUNfjKQ/mnoh55xHKb+VSUEwNuNPKzbRsMGwYvvwxjxkDZslZc5o474JBDglbnONExZoxNRc2dC2efbVOqXkHPiQNuNHJZu9YWt197DRYsgCpVrPpYmzawd+oE8zoZzi+/wN13Q9++dpMzeLBNRTlOnHCj8f33Vg/gnXfMcDRqZG6IF1wAcapn4TgJ5++/zSnjqadstPz445Zqv0yZoJU5aUZmGo2cHMuv07WrRcDusoutV7Rta0bDcVKJoUNt+nThQjj/fJtarVYtaFVOmpJZRmPZMnj7bXjjDRvGV65sacpvuMGKyjhOKrFwoRmLoUPh8MNh+HA466ygVTlpTvobjW3bYNQoMxSDB1ua8iZNzIX23HM9zYeTeqxbZzW5O3WymtzPPw+33+71uZ2kkL6/mL/8YusUPXta9s7y5a3y2A03QI0aQatznB1H1dbb7rkHfv0VrrjCnDUqVQpamZNBpJfR2LSJA8aMsSydo0bZl6xJE/tinX8+7Lpr0Aodp0SUW7AAHn0Uxo2zOKG+feHEE4OW5WQg6WM0cnKgenWO/PVXq7X98MNwzTUeW+GkPp9+Sv0bb4T99rNp1muv9RxnTmCkj9HYeWd4+GFmrl9P3Tvv9FxQTvpw5pn8dPnlVOvcGfbZJ2g1ToaTXr+sN93E6vr13WA46cXuu7P42mvdYDihwH9dHcdxnKhxo+E4juNEjYSlNsyOICIrgJ8K2V0eWJlEOYURFh3gWgqiKB0Hq2o0ReziThF9OyztBq6lIMKiAxLct1PSaBSFiExR1Qau4x9cS3h1REuY9LqW8OqAxGvx6SnHcRwnatxoOI7jOFGTjkbjjaAFRAiLDnAtBREWHdESJr2uZXvCogMSrCXt1jQcx3GcxJGOIw3HcRwnQbjRcBzHcaImZYyGiPQUkeUi8l0h+0VEXhWRhSIyS0Tq5dnXTETmR/bdn2Adl0Xef5aITBCRunn2LRaR2SIyQ0SmxKIjSi2NRWRN5P1miMijefbFrU2i1HJPHh3fichWEdk3si9u7SIiVUVkrIjMFZFsEbm9gGOS0lei1BtLf0r2Zximvt0yomOGiEwRkZPz7Evm970oHUltkzzHHRf5frXOsy1+fUVVU+IBnArUA74rZH8L4HNAgOOBSZHtpYAfgEOB0sBM4MgE6jgR2CfyvHmujsjrxUD5JLZJY+CzArbHtU2i0ZLv2HOBMYloF6ASUC/yfA/g+/zXlqy+ksj+FMRnGLK+XY5/1mSPBuYlol1KqiOINslz/WOAYUDrRLRJyow0VPUr4I8iDmkJ9FZjIrC3iFQCGgILVXWRqm4G+kSOTYgOVZ2gqqsjLycCVUr6XrFqKYK4tkkJtFwCfBTL+xWhY5mqTos8XwvMBSrnOywpfSVKvSXtT0n/DMPUt1V1nUZ+EYGyQO7zZH/fC9MRd6L8jrUF+gPL82yLa5ukjNGIgsrAL3leL4lsK2x7MrgOu6PNRYGRIjJVRNokScMJIjJTRD4XkdqRbYG1iYjsDjTDOnYuCWkXEakGHAtMyrcrjH0lGvL2p6C1Bt63RaSViMwDhgLXRjYnvV0K0QFJbhMRqQy0Arrn2xXXNkmfeho21ZAfLWJ7QhGR07Ev1sl5Np+kqktF5ABglIjMi9w9JIppWK6ZdSLSAhgE1CCgNolwLvC1qua9Y4p7u4hIOcww3aGqf+XfXcApgfWVaCigPwWmNSR9G1UdCAwUkVOBJ4EmBNAuheiA5LfJK8B9qrpV5F/NENc2SaeRxhKgap7XVYClRWxPGCJyNPAW0FJVV+VuV9Wlkb/LgYHYsDFhqOpfqrou8nwYsIuIlCeANsnDxeSbmop3u4jILpjB+EBVBxRwSGj6SjQU0p8C0RqWvp2XyA/xYUH37Xw6gmiTBkAfEVkMtAa6isj5xLtN4rVIk4wHUI3CF6TO5t+Lm99Gtu8MLAIO4Z9FoNoJ1HEQsBA4Md/2ssAeeZ5PAJoluE0q8s8iXUPg50j7xL1NitMS2b8XNidbNlHtErm+3sArRRyTtL6SwP6U9M8wZH27ep6+XQ/4NVF9u4Q6kt4m+Y57l38WwuPaJikzPSUiH2HeQOVFZAnQAdgFQFW7Y94CLbBOvQG4JrIvR0RuA0ZgXgQ9VTU7gToeBfbDrDxAjlrGyQrYEBbsQ/xQVYeXVEeUWloDN4tIDrARuFitF8W1TaLUAjbfOlJV1+c5Nd7tchJwBTBbRGZEtj2I/eAlta9EQ0n7UyK0pljfvhC4UkS2YH37okT07ZLqEJEg2qRA4v4bGLFEjuM4jlMs6bSm4TiO4yQYNxqO4zhO1LjRcJwiKC79ghjbpSQRkTIi8m0kRiZbRB5PvnrHiT8psxCel/Lly2u1atWClgHA+vXrKVu2bNAyEkKmXtvUqVNXqur+IlIKeB1oirktThaRIao6J8/hzbHYlxpAI6Bb5O/fwBlqMTK7AONF5HO1CPRCKaxvh+mzcC3h1QHR9e2Y3iBWF7AgHvXr19ewMHbs2KAlJIxMvTZgiv3hBGCE/uPG+ADwgObpi0AP4JI8r+cDlfIdszsWaNlIS9i3w/RZuJbtCYsO1ej6diyPlBxpOE6SKCj9QqMojqkMLIuMVKZivvyvq2r+dCYARFJMtAGoUKECWVlZ2x2zbt26ArcHgWsJrw5IvBY3Gk7m8ccf7DdhAjRuXNyR0aRfKPQYVd0KHCMie2M++3VUdbu01qr6BpESnQ0aNNDGBejKysqioO1B4FrCq4Px45m0Zg2NzjknYW/hC+FOZjFkCNSuzZFPPgl/FJuUN5r0C8Ueo6p/AllYokbHiT9r18Jtt8Epp1DtnXcS+lZuNJzMYNUquPxyaNkSKlRg+muvwb77FnfWZKCGiBwiIqWxvFlD8h0zBIsIFhE5HlijqstEZP/ICAMR2Q1LYjcvvhflOMCIEVCnDnTtCu3a8X379gl9OzcaTvozcCDUrg0ffwyPPQbffsu66tWLPU1Vc4Dc9Atzgb6qmi0iN4nITZHDhmF5fRYCbwK3RLZXAsaKyCzM+IxS1c/iel1OZvPHH3D11dCsGey+O4wfD507s3W33RL6tr6m4aQvK1dCu3bw0UdwzDF2R1a3bvHn5UEtO/CwfNu653muwK0FnDcLq+XhOPGnf3+49VYbQT/0EDz8MJQpk5S39pGGk57072+ji3794PHH4dtvd9hgOE7o+O03aN3aHgceCJMnw1NPJc1ggI80nHRjxQpbEOzbF+rVg9Gj4aijglblOLGhCr16wV13wYYN8OyzcPfdsHPyf8LjMtKIItVCYxFZIyIzIo9Hoz3XcaLmk09sdDFwIDz5JEyc6AbDSX0WL7Z1i2uusf49cybcf38gBgPiMNKIMtUCwDhVPaeE5zpO4SxfbvO7/fpBgwYwZox5kzhOKrNtG7z+OjzwAIhAly5w882wU7CrCvF494bAQlVdpKqbgT5AyySc62Q6qrbIfeSRFn/xzDPwzTduMJzUZ948OPVUc+Q45RT47ju7MQrYYEB8jEZhaRTyc0Ik4+fnIlJ7B891nH/z229w4YVw6aVQvTpMn253ZAEN2R0nLmzZYjc/devCnDm2jjFsGBx8cNDK/kc8vmHRpFqYBhyslvGzBTAIywoazbn2JlHk5wmCMOWciTehvDZVDhg9mhpdulBq40Z+vPFGfvnvf22KavnyqP9NKK/NyWymT4drr4UZM8w7qksXqFAhaFXbEQ+jEU0ahb/yPB8mIl1FpHw05+Y5r9j8PEEQmpwzCSB017Z0Kdx0E3z6KZxwAvTsyWFHHMFhJfhXobs2J3PZtAmeeAKefx7239/cxS+4IGhVhRKP6aliUy2ISEWJVFgXkYaR910VzbmO8z93w9q1YdQo6NQJxo2DI44IWpnjxMb48TYV9eyzcOWVNiUVYoMBcRhpqGqOiOSmWigF9MxNtRDZ3x1oDdwsIjnARuDiSCRtgefGqslJI375BW68ET7/3BYE334batQIWpXjxMbatfDgg+YddfDBMHIkNG0atKqoiMuqYRSpFroAXaI913FQhbfegvbtYetWePXV0HiPOE5MjBgBbdrYDVHbtvD001CuXNCqosZdTZzwsXgx3HCDRXM3bmyji0MPDVqV48TGH39YRHevXja1On48nHhi0Kp2GL9tc8LDtm2W3rlOHYvm7toVvvjCDYaT+vTrZ/FEH3xgCQanT09JgwE+0nDCwg8/wHXXwZdf2tzum2+GyjfdcUrEsmWWC23AAMuFNny4ZVxOYXyk4QTL1q3wyiuWI2r6dFvHGDHCDYaT2qjCu+/a6GLoUPOOmjQp5Q0G+EjDCZL58y2YacIEOPts6NEDKntCACfFWbzYFrpHjTKPvzffhMMPD1pV3PCRhpN8cnIskKluXZg7F3r3toA9NxhOKrNtm3n51aljOdC6dIGsrLQyGOAjDSfZfPedjS4mT4bzz4du3aBixaBVOU5M7P7zzzaqmDDB0pj36AEHHRS0rITgIw0nOWzZYhXG6tWDH3+0et0DBoTeYERRK0ZE5NXI/lkiUi+yvaqIjBWRuSKSLSK3J1+9k3AiCQYbXH+9Zabt3dsSDKapwQAfaTjJIG8itosvtiH8/vsHrapYoqz30hxLvlkDaAR0i/zNAdqr6jQR2QOYKiKjvFZMGjFtmnn8zZjBytNO44CPPw5lgsF44yMNJ3H8/bcVvD/uOEtlPmCA1b9IAYMRIZp6Ly2B3mpMBPYWkUqqukxVpwGo6lpgLp72Pz3YuNHS8Dds+L9+PeexxzLCYIAbDULUQ+MAABVMSURBVCdRTJpkU1FPPw2XXw7Z2dCqVdCqdpRo6r0Ue4yIVAOOBSbFXaGTXMaNM7fZjh3/STCYev06Jnx6yokvGzbAo4/Cyy/DgQfa/G7z5kGrKinR1Hsp8hgRKQf0B+7IWyLgX/8giloxYar/kYlaSm3YwKFvvknlQYPYWLEi37/wAqsbNLB63UnUEQ2J1uJGw4kfX31lc7wLF1pm2uefhz33DFpVLERT76XQY0RkF8xgfKCqAwp7k2hqxYSp/kfGaRk+3Gpz//IL3H47uz39NHXLlk2+jihJtBafnnJiZ+1aS5Vw2mkW4f3FF9C9e6obDIiu3ssQ4MqIF9XxwBpVXRapH/M2MFdVX0qubCcu/PEHXHWVjZTLloWvv7bsBfkMRqbhRsOJjVGjLAVI165w++0wezaccUbQquKCquYAufVe5gJ9c2vF5NaLwdL6LwIWAm8Ct0S2nwRcAZwhIjMijxbJvQKnRKhagsFateDDD82ZY/p0qxbp+PSUU0JWr7ZaF++8YxGvKZrmuTiiqBWjwK0FnDeegtc7nDCzbJnVbRk4EOrXt+JIdesGrSpU+EjD2XEGD7bSq717w/33W/xFGhoMJ4NQhZ49LcHgsGHw3HOWnt8Nxnb4SMOJnhUroF076NMHjj7a8kXVrx+0KseJjR9/tASDo0dbKpC33oKaNYNWFVriMtKIItXCZZEUC7NEZIKI1M2zb7GIzI7M+U6Jhx4nzqiaoTjySOjfHx5/3HJHucFwUpmtW6Fz538X/crKcoNRDDGPNKJMtfAjcJqqrhaR5ph7YaM8+09X1ZWxanESwNKlcMstNiXVsKGVXq1TJ2hVjhMbc+bA9ddbNtrmzS3BYNWqxZ/nxGWkUWyqBVWdoKqrIy8nYr7sTphRpeLQoTa6GDECXnzRMni6wXBSmdzEmcceC99/D++9Z0WS3GBETTzWNApKo9CokGMBrgM+z/NagZEiokCPSKDTdkQTNRsEYYoEjRdlfvuNmi++yBFTp/Ln0Ucz/5572FiliqVQSBPS8XNzimHqVAs+nTkT/u//4LXX4IADglaVcsTDaESTasEOFDkdMxon59l8kqouFZEDgFEiMk9Vv9ruH0YRNRsEYYoEjZlt26xwzIMPggjf33EHNTt1otFO6edkl1afm1M0GzfCY4/ZaLliRRg0CFrmzzvpREs8fg2iSbWAiBwNvAW0VNVVudtVdWnk73JgIDbd5SSbefPg1FMtQO+UUyA7m6UtW0IaGgwng/jqK3Obff55S8+fne0GI0bi8YtQbKoFETkIGABcoarf59leNlJrABEpC/wH+C4Ompxo2bLFit4fc4wtDvbqlfZFZJwM4K+/LF/UaadZeeHRo61W9957B60s5Yl5ekpVc0QkN9VCKaBnbqqFyP7uwKPAfkBXS8lDjqo2ACoAAyPbdgY+VNXhsWpyomT6dJvjnT4dLrzQpqZCXknPcYpl2DBLmLl0Kdx5Jzz5ZMbni4oncQnuiyLVwvXA9QWctwjwkMtks2mTfZGeew7Kl7c8OxdeGLQqx4mNlSvNSLz/vnn99esHjYryyXFKgkeEZxoTJtjoYt48y+D50kuw775Bq3KckqMKfftC27aWE61DB6ust+uuQStLS9xoZArr1sFDD5mbYdWqViPgrLOCVuU4MVF65UqrnDd4MDRoYGn5jzoqaFlpjRuNTGDUKMuts3ixZfB89lnYY4+gVTlOyVGFt9+m4R13WDqQ55+3qamd/Sct0XgLpzP505ePGwcnn1z8eY4TZhYtghtugDFjWFe3Lnv36wfVqwetKmNwo5GuDBhgo4oVK2x+99FHoUyZoFU5TsnZuhVefdWKIpUqBd27M6NGDRq7wUgqHrmVbvz2G7Rubd5QFStaNtpnnnGD4aQ22dlw0klw111w+ukWU3TjjR58GgDe4umCKrz7rrkafvaZrVt8+60lZnOcVGXzZnjiCevHP/xg5Vc//RSqeM7ToPDpqXRg8WK76xo50tYs3nrL1jAcJ5WZPNlSf3z3HVxyidW+2H//oFVlPD7SSGVy53jr1LH4iy5d4Msv3WDEkSgKjImIvBrZP0tE6uXZ11NElouIp8bZETZsgLvvhuOPN2eOIUNshOEGIxS40UhV5syxxIK3326JBrOzbeHb53jjRp4CY82BI4FLROTIfIc1B2pEHm2Abnn2vQs0S7zSNCIry0oJd+pkRZKys+Hcc4NW5eTBf2FSjc2bLQXIscfC/PnQu7cVkfEEg4mg2AJjkde91ZgI7C0ilQAiKf7/SKriVGXNGptiPf10ez1mjFXT22uvYHU52+FrGqnE5MmWAmT2bLjoIpua8iIyiSSaAmMFHVMZWBbtm0RTYCxMRaPirWW/CROo+fLLlP7jD3656CIWX30120Rs1JFkLSUlLDog8VrcaKQCGzZYnMXLL5sb7eDBcN55QavKBKIpMBZ1EbLCiKbAWJiKRsVNy4oVNr360Ue2LjdsGAcddxw7MmYOS7uERQckXotPT4WdMWMsl06nThYFO2eOG4zkEU2BsaiKkDl5ULWF7dxMtI89ZqVYjzsuaGVOFLjRCCt//mkLgWeeaYvbWVnQvbvP8SaXYguMRV5fGfGiOh5Yo6pRT01lHEuW2ML2ZZfBYYdZLZcOHaB06aCVOVHiRiOMDBwItWpZsN5998GsWVaBzEkqqpoD5BYYmwv0zS0wlltkDKsjswhYCLwJ3JJ7voh8BHwDHC4iS0TkuqReQJjYtg3eeANq14axYy0l/9df22snpfA1jTCxbBncdpvljTrmGPOKqlev+POchBFFgTEFbi3k3EsSqy5FWLDApla//BLOOMOMx2GHBa3KKSE+0ggDkTTP1KplhiI3BYgbDCeVycmBF16wuIsZM6xG9+jRbjBSnLgYjRijZos8N+354Qdo0sTWL+rWtamo+++HXXYJWpnjlJzZs+GEE+Dee6FZM3PguP56kIKczZxUImajEUvUbJTnpic5OfDii+YZNWWKLXKPHQs1awatzHFKzt9/28J2vXrw00/w8cc23XrggUErc+JEPNY0/hc1CyAiuVGzc/Ic87+oWWCiiORGzVaL4tz0Y+ZMC9KbOtXcZ7t2hcqVg1blOLExcaL16zlz4PLL4ZVXYL/9glblxJl4GI1YomajOReILmo2CHYk+nKnzZs5uFcvDurThy177cWCDh1YcdpptlC4YEFihZaAMEW5xpt0vraks3691Z9/9VVLWT5sGDRvHrQqJ0HEw2jEEjUbdTRtNFGzQRB19OVXX5ln1IIFcM01lH7xRWrvu2/C9cVCmKJc4006X1tS+eIL84z68Ue45Rbo2NHrz6c58VgIjyVqNv2jaXMTsZ12mq1jjBoFPXtCyA2G4xTJ6tU2FdWkCey8s90Uvf66G4wMIB5GI5ao2WjOTV0GD7ZUCW+9Be3bm0dJkyZBq3KcmCg/bpz16169LPh05kxL0+9kBDFPT6lqjojkRs2WAnrmRs1G9nfHgqNaYFGzG4Brijo3Vk2B89tv0Lat5dU5+mgYNMjz6jipz++/Q9u21PnkEw8+zWDiEhEeY9TsduemLKo29XT33bBxIzzzjD33mAsnlVGF99+HO+6AdetYdN11HNqtm/frDMUjwuPFwoWWXPD66210MWsWPPCAf7Gc1Oann6BFC7jySjjiCJgxg58vv9z7dQbjRiNGJCcHnnvOgvSmTrVqYx6k56Q627ZZzfnatWHcOHjtNftbq1bQypyA8YSFsTB1KvVuvtlGGa1a2RfLg/ScVGf+fPOM+vpr+M9/LMHgwQcHrcoJCT7SKAkbNsA990DDhpRevRr697dUCW4wnFRmyxaLs6hbF7KzLTX/8OFuMJx/4SONHWX0aGjTxoKZbriByeedx8nnnBO0KseJjenTbXQxfTpceKFNTVWsGLQqJ4T4SCNaVq2Cq6+Gpk0tmCkrC954g5xy5YJW5jglZ9MmePBBcwlftsxGzf36ucFwCsWNRnGoWuH7WrXggw/sC+aV9Jx0YPx4i7d49lnzjpozBy64IGhVTshxo1EUP/8M55wDl14K1aqZd9TTT0OZMkErc5ySs3at5UE75RRLZT5ypMUX7bNP0MqcFMCNRkFs3QqdO1uqhC+/hJdfhm++sfgLJ6NIuwJjw4ebG23XrtCunaW2ado0aFVOCuFGIz+zZsGJJ1r06ymnwHff2fNSpYJW5iSZtCowtmqVTUE1bw7lypk7befO9txxdgA3Grls2mQ1AerXN8+oDz6wugDVqgWtzAmO/xUYU9XNQG6RsLz8r8CYqk4EcguMRXNu4lGFTz6xUfNHH8HDD5uH1AknJF2Kkx64yy2YJ1SbNlbr4qqroFMnrzjmQIgKjJWkaFTplSup0bkz+48fz9qaNZn39NOsr17dplpjIEwFrMKiJSw6IPFaMttorF5the/fegsOPdRqXXjqcucfQlNgbIeKRuUmzmzf3ha6X3iBPe64g+N2js/XPUwFrMKiJSw6IPFaMtNo5A7Z27WDlSstuvuxx2D33YNW5oSLWAqMlY7i3PizaJEV/Ro9Gk491W6IatRI+Ns6mUPmrWn8/DOcdx5cdJHVM548GZ5/3g2GUxCpU2Bs61Z45RVLnDlpknlHjR3rBsOJO5kz0ti61cpRPvSQZfDs1MlGGnEasjvpR8oUGMvOthQgkybB2WdDt25QtWrx5zlOCciMX8zZs+GGG+xLddZZ0L27e0U5URHqAmObN1uCwaeegj33NI+/Sy4BKWg5xXHiQ3pPT23caGk/6tWzud7334fPP3eD4aQ+kyebe3iHDtC6Ncyda5kL3GA4CSYmoyEi+4rIKBFZEPm7XR4CEakqImNFZK6IZIvI7Xn2PSYiv4rIjMijRSx6/sWYMRbB/eyzcNll9qW67DL/UjmpzYYNVkL4+OPN+2/IEPjwQ9h//6CVORlCrCON+4EvVLUG8EXkdX5ygPaqWgs4Hrg1X2Tsy6p6TOQR+1B+1Sq49lorvQrmRfLuux534aQ+WVl2I9Spk023ZmfDuecGrcrJMGI1Gi2BXpHnvYDz8x+gqstUdVrk+VpgLhb4FF9U7Y6rVi147z2rzz1r1j/Gw3FSlTVrqNmpE5x+ur0eM8bW5fbaK1hdTkYS60J4hYh7Iaq6TEQOKOpgEakGHAtMyrP5NhG5EpiCjUhWF3JukVGzO23ezHH33MOW/fZj/jPPWOTrpEkF/Kf4EqZI0Hjj1xYSJk6k0rBhNi31+OPuHu4ESrFGQ0RGAwVVZHloR95IRMoB/YE7VPWvyOZuwJNYpOyTQCfg2oLOjyZqlgkT2K1KFY5LYnLBMEWCxhu/tpBw1llMeu89jr/00qCVOE7xRkNVC82rISK/i0ilyCijErC8kON2wQzGB6o6IM///j3PMW8Cn+2I+O3wWsZOmrLpwAODluA4QOxrGkOAqyLPrwIG5z9ARAR4G5irqi/l21cpz8tWwHcx6nEcx3ESSKxGoyPQVEQWAE0jrxGRA0Uk1xPqJOAK4IwCXGufF5HZIjILOB24M0Y9juM4TgIRC2hNLURkBfBT0DoilAdWBi0iQWTqtR2sqoEEPhTRt8P0WbiW7QmLDkhw305JoxEmRGSKqjYIWkci8GsLD2HS61rCqwMSryW904g4juM4ccWNhuM4jhM1bjRi542gBSQQv7bwECa9rmV7wqIDEqzF1zQcx3GcqPGRhuM4jhM1bjQcx3GcqHGjUQAi0lNElotIgRHqItJSRGZFAhWniMjJefbtLSL9RGRepIbICclTHh0xXt+dkboo34nIRyJSJnnKi6e4a8tz3HEislVEWufZ1kxE5ovIQhEpKM1/wnSVpE8VVY9GRB6IXMd8ETkrKC0iUk1ENubZ3j3P/0rI90xE2kauO1tEng+qTQrTkuw2EZGP87zXYhGZEU2bFIqq+iPfAzgVqAd8V8j+cvyzHnQ0MC/Pvl7A9ZHnpYG9g76eeF0fltL+R2C3yOu+wNVBX8+OXFvkmFLAGKwUa+s8234ADo18bjOBI4Nu86L6FPAYcHcB/+vIiP5dgUMi11UqIC3VinifROg4HRgN7Bp5fUCAbVKYlqS2Sb7zOwGPRtMmhT18pFEAqvoV8EcR+9dppNWBsliWXkRkT+xDfzty3GZV/TPBcneYkl5fhJ2B3URkZ2B3YGnChJaA4q4tQlssgWbeBJsNgYWqukhVNwN9sHoxSdEV5z7VEuijqn+r6o/AQuz6gtBSKAnScTPQUVX/juzL/YyDaJPCtBRKIj8bERHg/4CPIpuKbJPCcKNRQkSklYjMA4byTzr3Q4EVwDsiMl1E3hKRsoGJjIGCrk9VfwVeBH4GlgFrVHVkcCp3HBGpjCXH7J5vV2Xglzyvl5CIYmFFUMI+dVtkuqKn/FNuOeZriaMWgEMix38pIqckWEdN4BQRmRR5v+Mi24Nok8K0QHLbJJdTgN9VdUHkdYnaxI1GCVHVgap6BFat8MnI5p2xoWU3VT0WWE/BJXBDT0HXF/khaIkNZQ8EyorI5cGpLBGvAPep6tZ82wsqHp9Uf/QS9KluwGHAMZgR7xTZHvO1xFHLMuCgyPF3AR9G7ooTpWNnYB+stPQ9QN/IHXYQbVKYlmS3SS6X8M8oA0rYJm40YiQynDxMRMpjlnqJquaWDOyHfZApS77rawL8qKorVHULMAA4MVCBO04DoI+ILAZaA11F5Hzss6ua57gqBDT1Fm2fUtXfVXWrqm4D3uSfqYW4XUusWiJTH6siz6di8+Y1E6Ujsm+AGt8C27AEfklvk8K0BNAmRKaTLwA+zvMvStQmbjRKgIhUj9wxICL1sEWnVar6G/CLiBweOfRMYE5AMktMYdeHTUsdLyK7R/afidV8TxlU9RBVraaq1bAv1i2qOgiYDNQQkUNEpDRwMVYvJimUpE9J4fVohgAXi8iuInIIUAP4NggtIrK/iJSKPD80omVRonQAg4AzIufUjJyzMog2KUxLAG0CdsM3T1WX5NlWojaJtUZ4WiIiHwGNgfIisgToAOwCoKrdgQuBK0VkC7ARuCjP4lRb4IPID88i4Jokyy+WGK5vkoj0A6YBOcB0wpU+IZprKxBVzRGR24ARmCdVT1XNTqKukvSp50XkGGxKYTFwY+T/ZYtIX+xHIwe4Ne90XDK1YIuzT4hIDrAVuElV/0igjp5ATzGX1c3AVZFzgmiTArWISLLbBOwmKO/UVLH9pDA8jYjjOI4TNT495TiO40SNGw3HcRwnatxoOI7jOFHjRsNxHMeJGjcajuM4TtS40XAcx3Gixo2G4ziOEzX/D5YsKji4lsefAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "#import mpld3\n",
    "#mpld3.enable_notebook()\n",
    "import numpy as np \n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "f=lambda x:x**3+4*x**2-10\n",
    "\n",
    "t=np.linspace(1,2,100)\n",
    "plt.subplot(2,2,1)\n",
    "plt.plot(t,f(t),'r')\n",
    "plt.grid(True)\n",
    "##########################################\n",
    "t=np.linspace(1.2,1.4,100)\n",
    "plt.subplot(2,2,2)\n",
    "plt.plot(t,f(t),'r')\n",
    "plt.grid(True)\n",
    "##########################################\n",
    "t=np.linspace(1.35,1.4,100)\n",
    "plt.subplot(2,2,3)\n",
    "plt.plot(t,f(t),'r')\n",
    "plt.grid(True)\n",
    "#########################################\n",
    "t=np.linspace(1.365,1.367,100)\n",
    "plt.subplot(2,2,4)\n",
    "plt.plot(t,f(t),'r')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On considère l'équation :$$ f(x)=x^3+4x^2-10=0$$\n",
    "\n",
    "**a)** $g(x)=\\frac{1}{2}\\sqrt{10-x^3}=x$.\n",
    "\n",
    "$$x^3+4x^2-10=0$$\n",
    "\n",
    "$$4x^2=10-x^3$$\n",
    "\n",
    "$$2x=\\sqrt{10-x^3}$$\n",
    "\n",
    "$$x=\\frac{1}{2}\\sqrt{10-x^3}$$\n",
    "\n",
    "Donc $ f(x)=g(x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calcul symbolique de la dérivée"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "import numpy as np\n",
    "x, y, z = symbols('x y z')\n",
    "init_printing(use_unicode=True)\n",
    "g1=lambda x: 1/2*sqrt(10-x**3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -0.5117446282927461$"
      ],
      "text/plain": [
       "-0.5117446282927461"
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dg1=lambdify(x,diff(g1(x),x,1))\n",
    "dg1(1.365)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Que pouvez vous conclure à propos des convergences de g1?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**<font color='green'>Réponse:</font> g1 Converge localement car |g'(x)| < 1**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Justification théoriques pour g2 et g3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On considère l'équation :$$ f(x)=x^3+4x^2-10=0$$\n",
    "\n",
    "**b)** $g(x)=\\sqrt{\\displaystyle\\frac{10}{4+x}}=x$.\n",
    "\n",
    "$$x^3+4x^2-10=0$$$$x^3+4x^2=10$$$$x^2(x+4)=10$$$$x^2=\\frac{10}{x+4}$$$$x=\\sqrt{\\displaystyle\\frac{10}{4+x}}$$\n",
    "Donc $ f(x)=g(x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On considère l'équation :$$ f(x)=x^3+4x^2-10=0$$**c)** $g(x)=x-x^3-4x^2+10=x$.\n",
    "\n",
    "$$x^3+4x^2-10=0$$$$x^3+4x^2-10+x-x=0$$$$x^3-4x^2+x-x=10$$$$x^3-4x^2+x=x+10$$$$4x^2+x=x-x^3+10$$$$x=x-x^3-4x^2+10$$\n",
    "Donc $ f(x)=g(x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "f=lambda x:  x**3+4*x**2-10 # la fonction f\n",
    "g1=lambda x: 1/2*(math.sqrt(10-x**3))\n",
    "g2=lambda x: math.sqrt(10/(x+4))\n",
    "g3=lambda x: x-x**3-4*x**2 + 10\n",
    "\n",
    "\n",
    "def ptfixe(g,x0,eps,nmax):\n",
    "    zero=x0\n",
    "    for i in range(nmax):\n",
    "        x=zero \n",
    "        zero=g(x)\n",
    "        erreur=abs(zero-x)\n",
    "        print(zero)\n",
    "        if erreur<eps:\n",
    "            return zero,i\n",
    "    print('pas de convergence avec le nombre d''iteration indique')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.348399724926484\n",
      "1.3673763719912828\n",
      "1.364957015402487\n",
      "1.3652647481134421\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( 1.3652647481134421, \\  3\\right)$"
      ],
      "text/plain": [
       "(1.3652647481134421, 3)"
      ]
     },
     "execution_count": 130,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ptfixe(g2,1.5,0.001,50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.286953767623375\n",
      "1.4025408035395783\n",
      "1.3454583740232942\n",
      "1.3751702528160383\n",
      "1.360094192761733\n",
      "1.3678469675921328\n",
      "1.3638870038840212\n",
      "1.36591673339004\n",
      "1.364878217193677\n",
      "1.365410061169957\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( 1.365410061169957, \\  9\\right)$"
      ],
      "text/plain": [
       "(1.365410061169957, 9)"
      ]
     },
     "execution_count": 131,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "ptfixe(g1,1.5,0.001,50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-0.875\n",
      "6.732421875\n",
      "-469.72001200169325\n",
      "102754555.18738511\n",
      "-1.0849338705317464e+24\n",
      "pas de convergence avec le nombre diteration indique\n"
     ]
    }
   ],
   "source": [
    "ptfixe(g3,1.5,0.001,5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Quelle est la fonction qui donne la convergence la plus rapide?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**<font color='green'>Réponse:</font> G2 la fonction qui donne la convergence la plus rapide**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <u>Exercice 4:<u>(méthode de Newton)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1) On considère la fonction $f(x)=e^{-x}-x$. Ecrire puis appliquer une fonction Matlab que l'on appellera newton.m à la >fonction $f$, en prenant $x0=0$, $eps=0.001$ .\n",
    "\n",
    "2) Montrer que l'équation $f(x)=0$ admet une solution unique que l'on peut obtenir en utilisant la méthode de Newton à partir >de $x0=1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color='green'>Corrigé:</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import exp\n",
    "from time import time\n",
    "f=lambda x: exp(-x)-x\n",
    "fprime=lambda x: -exp(-x)-1\n",
    "def newton(a,eps):\n",
    "    debut=time()\n",
    "    x1=a\n",
    "    x2=x1-f(x1)/fprime(x1)\n",
    "    while (abs(x1-x2)>eps) :\n",
    "        x1=x2\n",
    "        x2=x1-(f(x1)/fprime(x1))\n",
    "        fin=time()\n",
    "    return x2,fin-debut"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle \\left( 0.567143285989123, \\  0.0\\right)$"
      ],
      "text/plain": [
       "(0.567143285989123, 0.0)"
      ]
     },
     "execution_count": 134,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newton(1,0.001)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "=> la méthode de newton est toujours convergente localement"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <font color='red'>Conclusion:</font>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Toute méthode devient le meilleur selon leur cas."
   ]
  }
 ],
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