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<div id="content">
<h1 class="title">À propos du calcul de π</h1>
<div id="table-of-contents">
<h2>Table des matières</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgd10738e">1. En demandant à la lib maths</a></li>
<li><a href="#org50a725d">2. En utilisant la méthode des aiguilles de Buffon</a></li>
<li><a href="#org7d95588">3. Avec un argument "fréquentiel" de surface</a></li>
</ul>
</div>
</div>
<div id="outline-container-orgd10738e" class="outline-2">
<h2 id="orgd10738e"><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 approximativement:
</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-org50a725d" class="outline-2">
<h2 id="org50a725d"><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 méthode des aiguilles de Buffon, on obtiendrait
comme approximation :
</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=pi/2)
<span style="color: #a0522d;">valeur</span> = 2/(<span style="color: #483d8b;">sum</span>((x+np.sin(theta))&gt;1)/N)
<span style="color: #a020f0;">print</span>(valeur)
</pre>
</div>
<pre class="example">
3.128911138923655
</pre>
</div>
</div>
<div id="outline-container-org7d95588" class="outline-2">
<h2 id="org7d95588"><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 méthode de Monte
Carlo sur Wikipedia). 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>
<div class="figure">
<p><img src="./valeurpip.png" alt="valeurpip.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"><span style="color: #a020f0;">print</span>(<span style="color: #8b2252;">'{:1.13f}'</span>.<span style="color: #483d8b;">format</span>(4*np.mean(accept)))
</pre>
</div>
<pre class="example">
3.1120000000000
</pre>
<p>
Auteur: Konrad Hinsen
</p>
<p>
Created: 2019-03-28 Thu 11:06
</p>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Auteur: Leraut Alain</p>
<p class="date">Created: 2020-04-20 lun. 18:01</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
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