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<body>
<div id="content">
<h1 class="title">À propos du calcul de Pi</h1>
<div id="table-of-contents">
<h2>Table des matières</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgbaf973c">1. En demandant à la lib maths</a></li>
<li><a href="#orgd8b2fe9">2. En utilisant la méthode des aiguilles de Buffon</a></li>
<li><a href="#org8726609">3. Avec un argument "fréquentiel" de surface</a></li>
</ul>
</div>
</div>
<div id="outline-container-orgbaf973c" class="outline-2">
<h2 id="orgbaf973c"><span class="section-number-2">1</span> En demandant à la lib maths</h2>
<div class="outline-text-2" id="text-1">
<p>
Mon ordinateur m'indique que π vaut <i>approximativement</i>:
</p>
<div class="org-src-container">
<pre class="src src-python"><span style="color: #a020f0;">from</span> math <span style="color: #a020f0;">import</span> *
<span style="color: #a020f0;">print</span>(pi)
</pre>
</div>
<pre class="example">
3.141592653589793
</pre>
</div>
</div>
<div id="outline-container-orgd8b2fe9" class="outline-2">
<h2 id="orgd8b2fe9"><span class="section-number-2">2</span> En utilisant la méthode des aiguilles de Buffon</h2>
<div class="outline-text-2" id="text-2">
<p>
Mais calculé avec la <b>méthode</b> des <a href="https://fr.wikipedia.org/wiki/Aiguille_de_Buffon">aiguilles de Buffon</a>, on obtiendrait comme <b>approximation</b> :
</p>
<div class="org-src-container">
<pre class="src src-python"><span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
np.random.seed(seed=42)
<span style="color: #a0522d;">N</span> = 10000
<span style="color: #a0522d;">x</span> = np.random.uniform(size=N, low=0, high=1)
<span style="color: #a0522d;">theta</span> = np.random.uniform(size=N, low=0, high=np.pi/2)
<span style="color: #a020f0;">print</span>(2/(<span style="color: #483d8b;">sum</span>((x+np.sin(theta))&gt;1)/N))
</pre>
</div>
<pre class="example">
3.128911138923655
</pre>
</div>
</div>
<div id="outline-container-org8726609" class="outline-2">
<h2 id="org8726609"><span class="section-number-2">3</span> Avec un argument "fréquentiel" de surface</h2>
<div class="outline-text-2" id="text-3">
<p>
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∼U(0,1) et Y∼U(0,1) alors P[X2+Y2≤1]=π/4 (voir
<a href="https://fr.wikipedia.org/wiki/M%C3%A9thode_de_Monte-Carlo#D%C3%A9termination_de_la_valeur_de_%CF%80">Méthode
de Monte Carlo sur Wikipédia</a>). Le code suivant illustre ce fait :
</p>
<div class="org-src-container">
<pre class="src src-python"><span style="color: #a020f0;">import</span> matplotlib.pyplot <span style="color: #a020f0;">as</span> plt
np.random.seed(seed=42)
<span style="color: #a0522d;">N</span> = 1000
<span style="color: #a0522d;">x</span> = np.random.uniform(size=N, low=0, high=1)
<span style="color: #a0522d;">y</span> = np.random.uniform(size=N, low=0, high=1)
<span style="color: #a0522d;">accept</span> = (x*x+y*y) &lt;= 1
<span style="color: #a0522d;">reject</span> = np.logical_not(accept)
<span style="color: #a0522d;">fig</span>, <span style="color: #a0522d;">ax</span> = plt.subplots(1)
ax.scatter(x[accept], y[accept], c=<span style="color: #8b2252;">'b'</span>, alpha=0.2, edgecolor=<span style="color: #008b8b;">None</span>)
ax.scatter(x[reject], y[reject], c=<span style="color: #8b2252;">'r'</span>, alpha=0.2, edgecolor=<span style="color: #008b8b;">None</span>)
ax.set_aspect(<span style="color: #8b2252;">'equal'</span>)
plt.savefig(matplot_lib_filename)
<span style="color: #a020f0;">print</span>(matplot_lib_filename)
</pre>
</div>
<p>
<a href="&lt;matplotlib.collections.PathCollection object at 0x7f4200f5fac0&gt;">&lt;matplotlib.collections.PathCollection object at 0x7f4200f5fac0&gt;</a>
</p>
<div class="figure">
<p><img src="./figure.png" alt="figure.png" />
</p>
</div>
<p>
Il est alors aisé d'obtenir une approximation (pas terrible) de π en
comptant combien de fois, en moyenne, X2+Y2 est inférieur à 1 :
</p>
<div class="org-src-container">
<pre class="src src-python">4*np.mean(accept)
</pre>
</div>
<pre class="example">
3.112
</pre>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="date">Date: La date du jour</p>
<p class="author">Auteur: Votre nom</p>
<p class="date">Created: 2020-04-23 jeu. 18:45</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
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