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zD&Y&`*K_yq_+Vz14}-z%{QYg<>$5KW5%{SsiHL%LPy@oC{l9*rkPr}bu~6|4 l5Wbr;JzYZb|I;NSCh#xT^=(Reqrj6P$V;n8RY{nJ{0}C^lo|j4 literal 0 HcmV?d00001 diff --git a/module2/exo1/toy_document_orgmode_python_fr.html b/module2/exo1/toy_document_orgmode_python_fr.html new file mode 100644 index 0000000..b5c7930 --- /dev/null +++ b/module2/exo1/toy_document_orgmode_python_fr.html @@ -0,0 +1,381 @@ + + + + + + + +À propos du calcul de π + + + + + + + + + + + + +

+

À propos du calcul de π

+ + +
+

1 En demandant à la lib maths

+
+

+Mon ordinateur m'indique que π vaut approximativement : +

+ +
+
from math import *
+pi
+
+
+ +
+3.141592653589793
+
+
+
+ +
+

2 En utilisant la méthode des aiguilles de Buffon

+
+

+Mais calculé avec la méthode des aiguilles de Buffon, on obtiendrait comme +approximation : +

+ +
+
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=pi/2)
+2/(sum((x+np.sin(theta))>1)/N)
+
+
+ +
+3.128911138923655
+
+
+
+ +
+

3 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 \(X \sim U(0,1)\) et \(Y \sim U(0,1)\) +alors \(P[X^2+Y^2 \leq1] = \pi/4\) (voir +méthode +yde Monte Carlo sur Wikipedia). Le code suivant illustre ce fait : +

+ +
+
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)
+matplot_lib_filename
+
+
+ +

+fige.png +Il est alors aisé d'obtenir une approximation (pas terrible) de π en comptant +combien de fois, en moyenne, \(X^2+Y^2\) est inférieur à 1 : +

+
+
4*np.mean(accept)
+
+
+ +
+3.112
+
+
+
+
+
+

Date: 01/04/2020

+

Auteur: Clément

+

Created: 2020-04-01 Wed 10:17

+

Validate

+
+ + diff --git a/module2/exo1/toy_document_orgmode_python_fr.org b/module2/exo1/toy_document_orgmode_python_fr.org index c7157ba..cf15efe 100644 --- a/module2/exo1/toy_document_orgmode_python_fr.org +++ b/module2/exo1/toy_document_orgmode_python_fr.org @@ -1,17 +1,82 @@ -#+TITLE: Votre titre -#+AUTHOR: Votre nom -#+DATE: La date du jour +#+TITLE: À propos du calcul de \pi +#+AUTHOR: Clément +#+DATE: 01/04/2020 #+LANGUAGE: fr # #+PROPERTY: header-args :eval never-export -#+HTML_HEAD: -#+HTML_HEAD: +# #+HTML_HEAD: +# #+HTML_HEAD: #+HTML_HEAD: #+HTML_HEAD: #+HTML_HEAD: #+HTML_HEAD: -* Quelques explications +* En demandant à la lib maths +Mon ordinateur m'indique que \pi vaut /approximativement/ : + +#+begin_src python :session :exports both +from math import * +pi +#+end_src + +#+RESULTS: +: 3.141592653589793 + +* En utilisant la méthode des aiguilles de Buffon +Mais calculé avec la *méthode* des [[https://fr.wikipedia.org/wiki/Aiguille_de_Buffon][aiguilles de Buffon]], on obtiendrait comme +*approximation* : + +#+begin_src python :session :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=pi/2) +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 \(X \sim U(0,1)\) et \(Y \sim U(0,1)\) +alors \(P[X^2+Y^2 \leq1] = \pi/4\) (voir +[[https://fr.wikipedia.org/wiki/M%C3%A9thode_de_Monte-Carlo#D%C3%A9termination_de_la_valeur_de_%CF%80][méthode +yde Monte Carlo]] sur Wikipedia). Le code suivant illustre ce fait : + +#+begin_src python :exports both :session :var matplot_lib_filename="fige.png" :results file +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) +matplot_lib_filename +#+end_src + +#+RESULTS: +[[file:fige.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 + +* Quelques explications :noexport: Ceci est un document org-mode avec quelques exemples de code python. Une fois ouvert dans emacs, ce document peut aisément être -- 2.18.1