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module2/EXO2/toy_document_orgmode_python_fr

+title: "On the computation of pi"		
+auteur : Meiling WU
+output: html_notebook
+#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="http://www.pirilampo.org/styles/readtheorg/css/htmlize.css"/>
+#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="http://www.pirilampo.org/styles/readtheorg/css/readtheorg.css"/>
+#+HTML_HEAD: <script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.3/jquery.min.js"></script>
+#+HTML_HEAD: <script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.4/js/bootstrap.min.js"></script>
+#+HTML_HEAD: <script type="text/javascript" src="http://www.pirilampo.org/styles/lib/js/jquery.stickytableheaders.js"></script>
+#+HTML_HEAD: <script type="text/javascript" src="http://www.pirilampo.org/styles/readtheorg/js/readtheorg.js"></script>
+
+#+PROPERTY: header-args  :session  :exports both
+
+* En demandant à la lib maths
+Mon ordinateur m'indique que π vaut /approximativement:
+#+begin_src python :results output :exports both
+from math import *
+print(pi)
+#+end_src
+
+#+RESULTS:
+: 3.141592653589793
+
+* En utilisant la méthode des aiguilles de Buffon
+Mais calculé avec la *méthode des aiguilles de Buffon, on obtiendrait comme *approximation :
+
+#+begin_src python :results output :exports both
+import numpy as np
+np.random.seed(seed=42)
+N = 10000
+x = np.random.uniform(size=N, low=0, high=1)
+theta = np.random.uniform(size=N, low=0, high=np.pi/2)
+print(2/(sum((x+np.sin(theta))>1)/N))
+#+end_src
+
+#+RESULTS:
+: 3.128911138923655
+
+* Avec un argument "fréquentiel" de surface
+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 $X\sim
+U(0,1)$, alors $P[X^2+Y^2\leq 1] = \pi/4$ (voir [[https://en.wikipedia.org/wiki/Monte_Carlo_method]["Monte Carlo method" on
+Wikipedia]]). Le code suivant illustre ce fait :
+
+#+begin_src python :results output :exports both
+import matplotlib.pyplot as plt
+import numpy as np
+
+matplot_lib_filename = "fig.png"
+np.random.seed(seed=42)
+N = 1000
+x = np.random.uniform(size=N, low=0, high=1)
+y = np.random.uniform(size=N, low=0, high=1)
+
+accept = (x*x+y*y) <= 1
+reject = np.logical_not(accept)
+
+fig, ax = plt.subplots(1)
+ax.scatter(x[accept], y[accept], c='b', alpha=0.2, edgecolor=None)
+ax.scatter(x[reject], y[reject], c='r', alpha=0.2, edgecolor=None)
+ax.set_aspect('equal')
+
+plt.savefig(matplot_lib_filename)
+print("./fig.png")
+#+end_src
+
+#+RESULTS:
+: ./fig.png
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