diff --git a/module2/exo1/toy_notebook_fr.ipynb b/module2/exo1/toy_notebook_fr.ipynb index 0bbbe371b01e359e381e43239412d77bf53fb1fb..1f84c6e5a22e84b84bb08511913dc7a6984d7baa 100644 --- a/module2/exo1/toy_notebook_fr.ipynb +++ b/module2/exo1/toy_notebook_fr.ipynb @@ -1,6 +1,146 @@ { - "cells": [], + "cells": [ + { + "cell_type": "markdown", + "metadata": { + "hideCode": false, + "hidePrompt": false + }, + "source": [ + "# 1 À propos du calcul de *$\\pi$* " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "hideCode": false, + "hidePrompt": false + }, + "source": [ + "## 1.1 En demandant à la lib maths " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "hideCode": false, + "hidePrompt": false + }, + "source": [ + "Mon ordinateur m’indique que π vaut *approximativement*" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "hideCode": false, + "hidePrompt": false + }, + "outputs": [], + "source": [ + "from math import *\n", + "print(pi)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "hideCode": false, + "hidePrompt": false + }, + "source": [ + "## 1.2 En utilisant la méthode des aiguilles de Buffon" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "hideCode": false, + "hidePrompt": false + }, + "source": [ + "Mais calculé avec la **méthode** des [aiguilles de Buffon](https://fr.wikipedia.org/wiki/Aiguille_de_Buffon), on obtiendrait comme **approximation** :" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "hideCode": false, + "hidePrompt": false + }, + "outputs": [], + "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": "markdown", + "metadata": { + "hideCode": false, + "hidePrompt": false + }, + "source": [ + "## 1.3 Avec un argument \"fréquentiel\" de surface" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "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^2 + Y^2 ≤ 1] = π/4 (voir\n", + "[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 :" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "import matplotlib.pyplot as plt\n", + "\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", + "\n", + "accept = (x*x+y*y) <= 1\n", + "reject = np.logical_not(accept)\n", + "\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')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + " Il est alors aisé d’obtenir une approximation (pas terrible) de $\\pi$ en comptant combien de fois,en moyenne, X^2+Y^2 est inférieur à 1 :" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "4*np.mean(accept)" + ] + } + ], "metadata": { + "hide_code_all_hidden": false, "kernelspec": { "display_name": "Python 3", "language": "python", @@ -16,10 +156,9 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.6.3" + "version": "3.6.4" } }, "nbformat": 4, "nbformat_minor": 2 } -