NM3_ex_solution.ipynb

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    "## Exercice : méthode de Romberg\n",
    "\n",
    "Estimation de \n",
    "\n",
    "$$\n",
    "\\int_0^{\\frac{3\\pi}{4}} \\sin x\\ dx\n",
    "$$\n",
    "\n",
    "Par la méthode de Romberg. Cette intégrale vaut $1 + 1/\\sqrt{2} \\approx 1.7071067811865475$"
   ]
  },
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   "cell_type": "code",
   "execution_count": 38,
   "id": "accomplished-coordinator",
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     "name": "stdout",
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     "text": [
      "R11=0.8330405509046936\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import math\n",
    "\n",
    "def f(x):\n",
    "    return np.sin(x)\n",
    "\n",
    "# méthode des trapèzes avec 2 points:\n",
    "x_0 = 0\n",
    "x_n = 3 * math.pi / 4\n",
    "R11 = (f(x_0) + f(x_n))/2 * (x_n - x_0)\n",
    "print(f\"{R11=}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "annoying-access",
   "metadata": {},
   "source": [
    "Calcul de R21 par la méthode itérative des trapèzes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "fewer-partition",
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   "outputs": [
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     "text": [
      "R21=1.5049402075046334\n"
     ]
    }
   ],
   "source": [
    "R21 = R11/2 + f(x_n/2) * (x_n - x_0)/2\n",
    "print(f\"{R21=}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "returning-mission",
   "metadata": {},
   "source": [
    "calcul de R22 par Richardson avec $p=2$ puisque l'erreur est dominée par $h^2$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "liable-forestry",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "R22=1.7289067597046133\n"
     ]
    }
   ],
   "source": [
    "R22 = (4*R21 - R11)/3\n",
    "print(f\"{R22=}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "streaming-tucson",
   "metadata": {},
   "source": [
    "Calcul de R31 par la méthode itérative des trapèzes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "cathedral-investigation",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "R31=1.6574582026781939\n"
     ]
    }
   ],
   "source": [
    "h = (x_n - x_0)/2 # espace entre les nouveaux points\n",
    "x = x_0 + h/2 # position x du premier point\n",
    "R31 = R21/2 + (f(x) + f(x+h)) * h/2\n",
    "print(f\"{R31=}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "sacred-pension",
   "metadata": {},
   "source": [
    "calcul de R32 par Richardson avec $p=2$ puisque l'erreur sur R21 et R31 est dominée par $h^2$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "compliant-zimbabwe",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "R32=1.708297534402714\n"
     ]
    }
   ],
   "source": [
    "R32 = (4*R31 - R21)/3\n",
    "print(f\"{R32=}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "elect-desktop",
   "metadata": {},
   "source": [
    "calcul de R33 par Richardson avec $p=4$ puisque l'erreur sur R22 et R32 est dominée par $h^4$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "floppy-alignment",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "R33=1.706923586049254\n"
     ]
    }
   ],
   "source": [
    "R33 = (16*R32 - R22)/15\n",
    "print(f\"{R33=}\")"
   ]
  },
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   "cell_type": "markdown",
   "id": "surgical-syndrome",
   "metadata": {},
   "source": [
    "Même chose en utilisant les fonctions définies au cours:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "green-league",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "R11=0.8330405509046936\n",
      "R21=1.5049402075046334\n",
      "R22=1.7289067597046133\n",
      "R31=1.6574582026781939\n",
      "R32=1.708297534402714\n",
      "R33=1.706923586049254\n",
      "R41=1.6947487165588795\n",
      "R42=1.7071788878524412\n",
      "R43=1.7071043114157562\n",
      "R44=1.7071071800723674\n",
      "\n",
      "En utilisant la formule analytique:\n",
      "INT=1.7071067811865475\n"
     ]
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from math import sin, pi, sqrt\n",
    "def trapezes_rec(f, x_0, x_n, i_old, k):\n",
    "    if k == 1: \n",
    "        return (f(x_0) + f(x_n))*(x_n - x_0)/2\n",
    "    else:\n",
    "        n = 2**(k-2)\n",
    "        h = (x_n - x_0)/n\n",
    "        x = x_0 + h/2\n",
    "        sum = 0\n",
    "        for i in range(n):\n",
    "            sum += f(x)\n",
    "            x += h\n",
    "        return i_old/2 + h*sum/2\n",
    "\n",
    "    \n",
    "def richardson(i_1, i_2, k):\n",
    "    fact = 2**k\n",
    "    return (fact*i_2 - i_1)/(fact - 1)\n",
    "\n",
    "# ligne 1\n",
    "R11 = trapezes_rec(np.sin, 0, 3*pi/4, 0, 1)\n",
    "print(f\"{R11=}\")\n",
    "\n",
    "# ligne 2\n",
    "R21 = trapezes_rec(np.sin, 0, 3*pi/4, R11, 2)\n",
    "print(f\"{R21=}\")\n",
    "R22 = richardson(R11, R21, 2)\n",
    "print(f\"{R22=}\")\n",
    "\n",
    "# ligne 3\n",
    "R31 = trapezes_rec(np.sin, 0, 3*pi/4, R21, 3)\n",
    "print(f\"{R31=}\")\n",
    "R32 = richardson(R21, R31, 2)\n",
    "print(f\"{R32=}\")\n",
    "R33 = richardson(R22, R32, 4)\n",
    "print(f\"{R33=}\")\n",
    "\n",
    "# ligne 4\n",
    "R41 = trapezes_rec(np.sin, 0, 3*pi/4, R31, 4)\n",
    "print(f\"{R41=}\")\n",
    "R42 = richardson(R31, R41, 2)\n",
    "print(f\"{R42=}\")\n",
    "R43 = richardson(R32, R42, 4)\n",
    "print(f\"{R43=}\")\n",
    "R44 = richardson(R33, R43, 6)\n",
    "print(f\"{R44=}\")\n",
    "\n",
    "print(\"\\nEn utilisant la formule analytique:\")\n",
    "print('INT={}'.format(1+1/math.sqrt(2)))"
   ]
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