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zI5y3XJXX4AWhL*BAB5htxHbbY*O)NP@F>`=`99t*6I?C_1khy9;ez!*!s`$@H>Yd7zT=nk;s_#Brtr4^_2ne{x`*&I2 z?;jjk98h#&HgBPsa-qFSR}i{d30d2p`}hBM&(&J>bYX~A`3!?sL-!Vte1gEu`L6+j z>%tsJ4;a1$43{tSQNXQ&NXecLIFCdcneJkR^zNQ$0?~)-rrcZpV-3-BjkUF&Xfs#W zy3C;lGB6Yw3O`Ln6_o(R&@fCh8yk64G_=L~b(P5Z8#54JN&0Ebs`kHPUkbY2n{yFD z<0MS?75{N(e0x9ect!E}Ix{mOv~Q?$N0@;qa~o7qQIP@c za*4>uwEkJ8QGI`g92e7ZB`2jU`B}mw G_1)/N) +#+end_src + +#+RESULTS: + +* 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 basse sur le fait que si $$ X \sim U(0,1)$$ et $$ Y \sim U(0,1) $$ alors $$ P[ X^2 + Y^2 \leq 1 ] = \frac{\pi}{4} $$ ( voir [[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 Wikipedia]]). Le code suivant illustre ce fait : + +#+begin_src python :results file :session :var matplot_lib_filename="./figure.png" :exports both +import matplotlib.pyplot as plt + +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(matplot_lib_filename) +#+end_src + +#+RESULTS: +[[file:figure.png]] + +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 : + +#+begin_src python :results output :session :exports both +4*np.mean(accept) +#+end_src + +#+RESULTS: +: 3.112 -- 2.18.1