Solving exo1

module2/exo1/toy_document_orgmode_python_en.org

-#+TITLE:  Your title
-#+AUTHOR: Your name
-#+DATE:   Today's date
+#+TITLE:  On the computation of pi
+#+AUTHOR: Laudin Molina
+#+DATE:   2020-09-11 
 #+LANGUAGE: en
-# #+PROPERTY: header-args :eval never-export
+#+PROPERTY: header-args :session :exports both
 
 #+HTML_HEAD: <link rel="stylesheet" type="text/css" href="http://www.pirilampo.org/styles/readtheorg/css/htmlize.css"/>
 #+HTML_HEAD: <link rel="stylesheet" type="text/css" href="http://www.pirilampo.org/styles/readtheorg/css/readtheorg.css"/>
@@ -11,84 +11,72 @@
 #+HTML_HEAD: <script type="text/javascript" src="http://www.pirilampo.org/styles/lib/js/jquery.stickytableheaders.js"></script>
 #+HTML_HEAD: <script type="text/javascript" src="http://www.pirilampo.org/styles/readtheorg/js/readtheorg.js"></script>
 
-* Some explanations
+* Asking the math library
 
-This is an org-mode document with code examples in R.  Once opened in
-Emacs, this document can easily be exported to HTML, PDF, and Office
-formats. For more information on org-mode, see
-https://orgmode.org/guide/.
+My computer tells me that \pi is /approximatively/
 
-When you type the shortcut =C-c C-e h o=, this document will be
-exported as HTML. All the code in it will be re-executed, and the
-results will be retrieved and included into the exported document. If
-you do not want to re-execute all code each time, you can delete the #
-and the space before ~#+PROPERTY:~ in the header of this document.
-
-Like we showed in the video, Python code is included as follows (and
-is exxecuted by typing ~C-c C-c~):
-
-#+begin_src python :results output :exports both
-print("Hello world!")
+#+begin_src python :results value :session *python* :exports both
+from math import *
+pi
 #+end_src
 
 #+RESULTS:
-: Hello world!
+: 3.141592653589793
 
-And now the same but in an Python session. With a session, Python's
-state, i.e. the values of all the variables, remains persistent from
-one code block to the next. The code is still executed using ~C-c
-C-c~.
+* Buffon's needle
 
-#+begin_src python :results output :session :exports both
-import numpy
-x=numpy.linspace(-15,15)
-print(x)
+Applying the method of [[https://en.wikipedia.org/wiki/Buffon%27s_needle_problem][Buffon's needle]], we get the *approximation*
+
+#+begin_src python :results value :session *python* :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:
-#+begin_example
-[-15.         -14.3877551  -13.7755102  -13.16326531 -12.55102041
- -11.93877551 -11.32653061 -10.71428571 -10.10204082  -9.48979592
-  -8.87755102  -8.26530612  -7.65306122  -7.04081633  -6.42857143
-  -5.81632653  -5.20408163  -4.59183673  -3.97959184  -3.36734694
-  -2.75510204  -2.14285714  -1.53061224  -0.91836735  -0.30612245
-   0.30612245   0.91836735   1.53061224   2.14285714   2.75510204
-   3.36734694   3.97959184   4.59183673   5.20408163   5.81632653
-   6.42857143   7.04081633   7.65306122   8.26530612   8.87755102
-   9.48979592  10.10204082  10.71428571  11.32653061  11.93877551
-  12.55102041  13.16326531  13.7755102   14.3877551   15.        ]
-#+end_example
-
-Finally, an example for graphical output:
-#+begin_src python :results output file :session :var matplot_lib_filename="./cosxsx.png" :exports results
+: 3.128911138923655
+
+* Using a surface fraction argument
+
+A method that is easier to understand and does not make use of the $\sin$
+function is based on the fact that if $X\sim U(0,1)$ and $Y\sim U(0,1)$, then
+$P[X^2+Y^2 \leq 1]=\pi/4$ (see [[https://en.wikipedia.org/wiki/Monte_Carlo_method]["Monte Carlo method" on Wikipedia]]). The following
+code uses this approach:
+
+#+begin_src python :results output file :var matplot_lib_filename="figure_pi_mc2.png" :exports both :session *python* 
 import matplotlib.pyplot as plt
 
-plt.figure(figsize=(10,5))
-plt.plot(x,numpy.cos(x)/x)
-plt.tight_layout()
+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:./cosxsx.png]]
-
-Note the parameter ~:exports results~, which indicates that the code
-will not appear in the exported document. We recommend that in the
-context of this MOOC, you always leave this parameter setting as
-~:exports both~, because we want your analyses to be perfectly
-transparent and reproducible.
-
-Watch out: the figure generated by the code block is /not/ stored in
-the org document. It's a plain file, here named ~cosxsx.png~. You have
-to commit it explicitly if you want your analysis to be legible and
-understandable on GitLab.
-
-Finally, don't forget that we provide in the resource section of this
-MOOC a configuration with a few keyboard shortcuts that allow you to
-quickly create code blocks in Python by typing ~<p~, ~<P~ or ~<PP~
-followed by ~Tab~.
-
-Now it's your turn! You can delete all this information and replace it
-by your computational document.
+[[file:figure_pi_mc2.png]]
+
+It is then straightforward to obtain a (not really good) approximation to $\pi$ by
+counting how many times, on average, $X^2+Y^2$ is smaller than 1:
+
+
+#+begin_src python :results value :session *python* :exports both
+4*np.mean(accept)
+#+end_src
+
+#+RESULTS:
+: 3.112
+