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Mon_toy_notebook_fr
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Martha HENRI
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19/09/2023
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1.1 En demandant à la lib maths

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Mon ordinateur m'indique que $\pi$ vaut aproximativement

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In [1]:
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from math import *
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3.141592653589793
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1.2 En utilisant la méthode des aiguilles de Buffon

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

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In [2]:
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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)
+
+ +
+
+
+ +
+
+ + +
+ +
Out[2]:
+ + + + +
+
3.128911138923655
+
+ +
+ +
+
+ +
+
+
+
+

1.3 Avec un argument "frequentiel" 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 ~ U(0, 1) et Y ~ U(0, 1) alors P[X2 + Y2 <= 1] = $\pi$/4 (voir +méthode de Monte Carlo sur Wikipedia). Le code suivant illustre ce fait :

+ +
+
+
+
+
+
In [3]:
+
+
+
%matplotlib inline
+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')
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+ +
+ +
+ +
+
+ +
+
+
+
+

Il est alors aisé d’obtenir une approximation (pas terrible) de $\pi$ en comptant combien de fois, +en moyenne, X2 + Y2 est inférieur à 1 :

+ +
+
+
+
+
+
In [4]:
+
+
+
4*np.mean(accept)
+
+ +
+
+
+ +
+
+ + +
+ +
Out[4]:
+ + + + +
+
3.112
+
+ +
+ +
+
+ +
+
+
+ + + + + + diff --git a/module2/exo1/Mon_toy_notebook_fr.md b/module2/exo1/Mon_toy_notebook_fr.md new file mode 100644 index 0000000..1306fcc --- /dev/null +++ b/module2/exo1/Mon_toy_notebook_fr.md @@ -0,0 +1,87 @@ +
Mon_toy_notebook_fr
+ +
Martha HENRI
+ +
19/09/2023
+ + +## 1 A propos du calcul de $\pi$ + +## 1.1 En demandant à la lib maths + +Mon ordinateur m'indique que $\pi$ vaut *aproximativement* + + +```python +from math import * +print(pi) +``` + + 3.141592653589793 + + +## 1.2 En utilisant la méthode des aiguilles de Buffon + +Mais calculé avec la **méthode** des [aiguilles de Buffon](https://fr.wikipedia.org/wiki/Aiguille_de_Buffon), on obtiendrait comme __aproximation__ : + + + +```python +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 + + + +## 1.3 Avec un argument "frequentiel" 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 ~ U(0, 1) et Y ~ U(0, 1) alors P[X2 + Y2 <= 1] = $\pi$/4 (voir +[méthode de Monte Carlo sur Wikipedia](https://fr.wikipedia.org/wiki/M%C3%A9thode_de_Monte-Carlo)). Le code suivant illustre ce fait : + + +```python +%matplotlib inline +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') +``` + + +![png](output_12_0.png) + + +Il est alors aisé d’obtenir une approximation (pas terrible) de $\pi$ en comptant combien de fois, +en moyenne, X2 + Y2 est inférieur à 1 : + + +```python +4*np.mean(accept) +``` + + + + + 3.112 + + diff --git a/module2/exo1/output_12_0.png b/module2/exo1/output_12_0.png new file mode 100644 index 0000000000000000000000000000000000000000..8fe904e49d04efafe673c13a233a86fa12403c60 GIT binary patch literal 38079 zcmaHSbyOQ))NODnUJ4YK;O-vWwYXC#QrumOySux)ySqEZ-5rYCm)~3KTko&;vND+@ 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