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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/. - -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~): +* Asking the math library +My computer tells me that \pi is approximatively +#+begin_src python :results value :session *python* :exports both +from math import * +pi +#+end_src -#+begin_src python :results output :exports both -print("Hello world!") +#+RESULTS: +: 3.141592653589793 + +* Buffon's needle + 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: -: Hello world! +: 3.128911138923655 -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~. +* 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 $P[X^2+Y^2 \le 1]=\pi/4$ (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 :session :exports both -import numpy -x=numpy.linspace(-15,15) -print(x) -#+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 + + +#+begin_src python :results output :session *python* :var matplot_lib_filename="figure.png" :exports code 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 ~