4.3

parent c00a86b2
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"cell_type": "markdown", "cell_type": "markdown",
"metadata": {}, "metadata": {},
"source": [ "source": [
" 1 # À propos du calcul de $\\pi$\n", "# À propos du calcul de $\\pi$\n",
"\n", "\n",
" 1 ## 1.1 En demandant à la lib maths\n", "## En demandant à la lib maths\n",
"\n", "Mon ordinateur m'indique que $\\pi$ vaut *approximativement*\n",
"2 Mon ordinateur m'indique que $\\pi$ vaut *approximativement*\n",
"\n", "\n",
"\n", "\n",
"In [1]:\n", "In [1]:\n",
" 1 from math import *\n", "from math import *\n",
" 2 print (pi)\n", "print (pi)\n",
" \n", " \n",
"3.141592653589793\n", "3.141592653589793\n",
" \n", " \n",
"\n", "\n",
"## 1.2 En utilisant la méthode des aiguilles de Buffon\n", "## En utilisant la méthode des aiguilles de Buffon\n",
"\n", "Mais calculé avec la **méthode** des [aiguilles de Buffon](https://fr.wikipedia.org/wiki/Aiguille_de_Buffon), on \n",
"Mais calculé avec la **méthode** des [aiguilles de Buffon]\n",
"(https://fr.wikipedia.org/wiki/Aiguille_de_Buffon), on \n",
"obtiendrait comme **approximation** :\n", "obtiendrait comme **approximation** :\n",
"\n", "\n",
"\n",
"In [2]:\n", "In [2]:\n",
"1 import numpy as np\n", "import numpy as np\n",
"2 np.random.seed(seed=42)\n", "np.random.seed(seed=42)\n",
"3 N = 10000\n", "N = 10000\n",
"4 x = np.random.uniform(size=N, low=0, high=1)\n", "x = np.random.uniform(size=N, low=0, high=1)\n",
"5 theta = np.random.uniform(size=N, low=0, high=pi/2)\n", "theta = np.random.uniform(size=N, low=0, high=pi/2)\n",
"6 2/(sum((x+np.sin(theta))>1)/N)\n", "2/(sum((x+np.sin(theta))>1)/N)\n",
" \n", " \n",
"3.1289111389236548\n", "3.1289111389236548\n",
"\n", "\n",
"1 ## Avec un argument \"fréquentiel\" de surface\n", "## Avec un argument \"fréquentiel\" de surface\n",
"2 Sinon, une méthode plus simple à comprendre et ne faisant pas intervenir d'appel à la fonction sinus se base sur le fait que si $X\\sim U(0,1)$ et $Y\\sim U(0,1)$ alors $P[X^2+Y^2\\leq\n", "Sinon, une méthode plus simple à comprendre et ne faisant pas intervenir d'appel à la fonction sinus se base sur le fait que si $X\\sim U(0,1)$ et $Y\\sim U(0,1)$ alors $P[X^2+Y^2\\leq1] = \\pi/4$ (voir [méthode de Monte Carlo sur Wikipedia](https://fr.wikipedia.org/wiki/M%C3%A9thode_de_Monte-Carlo#D%C3%A9termination_de_la_valeur_de_%CF%80)). Le code suivant illustre ce fait :\n",
"1] = \\pi/4$ (voir [méthode de Monte Carlo sur Wikipedia]\n", "\n",
"(https://fr.wikipedia.org/wiki/M%C3%A9thode_de_Monte-Carlo#D%C3%A9termination_de_la_valeur_de_%CF%80)). Le code suivant illustre ce fait :\n",
"\n", "\n",
"In [3]:\n", "In [3]:\n",
"\n", "\n",
"1%matplotlib inline\n", "%matplotlib inline\n",
"2 iport matplotlib.pyplot as plt\n", "import matplotlib.pyplot as plt\n",
"3\n", "\n",
"4 np.random.dees(seed=42)\n", "np.random.seed(seed=42)\n",
"5 N = 1000\n", "N = 1000\n",
"6 x = np.random.uniform(size=N, low=0, high=1)\n", "x = np.random.uniform(size=N, low=0, high=1)\n",
"7 y = np.random.uniform(size=N, low=0, high=1)\n", "y = np.random.uniform(size=N, low=0, high=1)\n",
"8\n", "\n",
"9 accept = (x*x+y*y) <=1\n", "accept = (x*x+y*y) <=1\n",
"10 reject = np.logical_not'accept)\n", "reject = np.logical_not(accept)\n",
"11\n", "\n",
"12 fig, ax = plt.subplots(1)\n", "fig, ax = plt.subplots(1)\n",
"13 ax.scatter(x[accept], y [accept], c='b', alpha=0.2, edgecolor=None)\n", "ax.scatter(x[accept], y [accept], c='b', alpha=0.2, edgecolor=None)\n",
"14 ax.scatter(x[reject], y [reject], c='r', alpha=0.2, edgecolor=None)\n", "ax.scatter(x[reject], y [reject], c='r', alpha=0.2, edgecolor=None)\n",
"15 ax.set_aspect('equal')\n", "ax.set_aspect('equal')\n",
"\n" "\n"
] ]
} }
......
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