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On the computation of pi

-
    -
  1. Asking the math library + +
    +

    1 1 Asking the math library

    +
    +

    My computer tells me that π is approximatively -#beginsrc python :results output :exports both -from math import * +

    + +
    +
    from math import *
     pi
    -#endsrc
  2. + +
+ + -
  • * Buffon’s needle +
    +

    2 2 * Buffon’s needle

    +
    +

    Applying the method of Buffon’s needle, we get the approximation -#beginsrc python :results output :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) -#endsrc

  • -
  • -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 \(X \approx U(0,1)\) and \(Y \approx U(0,1)\), then \(P[X^2 + Y^2 \leq 1] = \pi/4\) (see “Monte Carlo method on Wikipedia”). The following code uses this approach: - #beginsrc python :results output :exports both -import matplotlib.pyplot as plt

    +
    +
    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 3. Using a surface fraction argument

    +

    -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) +A method that is easier to understand and does not make use of the sin function is based on the fact that if \(X \approx U(0,1)\) and \(Y \approx U(0,1)\), then \(P[X^2 + Y^2 \leq 1] = \pi/4\) (see “Monte Carlo method on Wikipedia”). The following code uses this approach:

    +
    +
    import matplotlib.pyplot as plt
     
    -

    -accept = (x*x+y*y) <= 1 -reject = np.logicalnot(accept) -

    +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) -

    -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.setaspect(’equal’) -

    +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') +
    +
    + +
    +matplot_lib_filename
    +
    -

    -plt.savefig(matplotlibfilename) -print(matplotlibfilename) -#endsrc -

    It is then straightforward to obtain a (not really good) approximation to π by counting how many times, on average, \(X^2 + Y^2\) is smaller than 1: -#beginsrc python :results output :exports both -4*np.mean(accept) -#endsrc -

  • - +

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

    Author: Victor Martins Gomes

    -

    Created: 2020-04-27 lun. 20:01

    +

    Created: 2020-04-28 mar. 11:29

    diff --git a/module2/exo1/toy_document_orgmode_python_en.org b/module2/exo1/toy_document_orgmode_python_en.org index e903cb0..9ff0230 100644 --- a/module2/exo1/toy_document_orgmode_python_en.org +++ b/module2/exo1/toy_document_orgmode_python_en.org @@ -1,46 +1,68 @@ #+TITLE: On the computation of pi #+LANGUAGE: en -# #+PROPERTY: header-args :eval never-export -1. Asking the math library - My computer tells me that \pi is approximatively - #begin_src python :results output :exports both - from math import * - pi - #end_src - -2. * 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 output :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 -3. 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 $X \approx U(0,1)$ and $Y \approx U(0,1)$, 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 :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 - - 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 output :exports both - 4*np.mean(accept) - #end_src +# #+PROPERTY: header-args :session :exports both +* 1 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 + +#+RESULTS: +: 3.141592653589793 + +* 2 * 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: +: 3.128911138923655 + +* 3. 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 $X \approx U(0,1)$ and $Y \approx U(0,1)$, 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" :session *python* :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:Traceback (most recent call last): + File "", line 1, in + File "/tmp/babel-ekv1Vl/python-2ZIXqL", line 4, in + np.random.seed(seed=42) + NameError: name 'np' is not defined]] + File "", line 1, in + File "/tmp/babel-ekv1Vl/python-xlXPMX", line 4, in + np.random.seed(seed=42) + NameError: name 'np' is not defined]] + +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 output :session *python* :exports both +4*np.mean(accept) +#+end_src + + #+RESULTS: + : 3.112 -- 2.18.1