diff --git a/module2/exo1/toy_notebook_fr.ipynb b/module2/exo1/toy_notebook_fr.ipynb index eb108e1a36d89055f8abc57c4afe1ca7655c9555..3dd6649030af2ad9538342e7259396815252d3eb 100644 --- a/module2/exo1/toy_notebook_fr.ipynb +++ b/module2/exo1/toy_notebook_fr.ipynb @@ -1,5 +1,12 @@ { "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "March 28, 2019" + ] + }, { "cell_type": "markdown", "metadata": {}, @@ -7,6 +14,13 @@ "# Premier test jupyter" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# À propos du calcul de π $\\pi$" + ] + }, { "cell_type": "code", "execution_count": 1, @@ -24,6 +38,128 @@ "import os\n", "print('hello world')" ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## En demandant à la lib maths\n", + "Mon ordinateur m’indique que π vaut approximativement" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "3.141592653589793\n" + ] + } + ], + "source": [ + "from math import *\n", + "print(pi)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# En utilisant la méthode des aiguilles de Buffon\n", + "Mais calculé avec la **méthode** des [aiguilles de Buffon](https://fr.wikipedia.org/wiki/Aiguille_de_Buffon), on obtiendrait comme **approximation** :\n" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "3.128911138923655" + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "import numpy as np\n", + "np.random.seed(seed=42)\n", + "N = 10000\n", + "x = np.random.uniform(size=N, low=0, high=1)\n", + "theta = np.random.uniform(size=N, low=0, high=pi/2)\n", + "2/(sum((x+np.sin(theta))>1)/N)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "ut[2]: 3.1289111389236548\n", + "1.3 Avec un argument \"fréquentiel\" de surface\n", + "Sinon, une méthode plus simple à comprendre et ne faisant pas intervenir d’appel à la fonction\n", + "sinus se base sur le fait que si X ∼ U(0, 1) et Y ∼ U(0, 1) alors P[X\n", + "2 + Y\n", + "2 ≤ 1] = π/4 (voir\n", + "méthode de Monte Carlo sur Wikipedia). Le code suivant illustre ce fait :\n", + "In [3]: %matplotlib inline\n", + "import matplotlib.pyplot as plt\n", + "np.random.seed(seed=42)\n", + "N = 1000\n", + "x = np.random.uniform(size=N, low=0, high=1)\n", + "y = np.random.uniform(size=N, low=0, high=1)\n", + "1\n", + "accept = (x*x+y*y) <= 1\n", + "reject = np.logical_not(accept)\n", + "fig, ax = plt.subplots(1)\n", + "ax.scatter(x[accept], y[accept], c='b', alpha=0.2, edgecolor=None)\n", + "ax.scatter(x[reject], y[reject], c='r', alpha=0.2, edgecolor=None)\n", + "ax.set_aspect('equal')\n", + "Il est alors aisé d’obtenir une approximation (pas terrible) de π en comptant combien de fois,\n", + "en moyenne, X\n", + "2 + Y\n", + "2\n", + "est inférieur à 1 :\n", + "In [4]: 4*np.mean(accept)\n", + "Out[4]: 3.1120000000000001\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] } ], "metadata": {