diff --git a/module2/exo1/toy_document_orgmode_R_en.org b/module2/exo1/toy_document_orgmode_R_en.org index 6a02df3bd9edac8b13ac9d738e3cc9b69f2fe5d4..0febcfe34f204f56d08923d6106792bf2520606a 100644 --- a/module2/exo1/toy_document_orgmode_R_en.org +++ b/module2/exo1/toy_document_orgmode_R_en.org @@ -1,4 +1,4 @@ -#+TITLE: On the computation of pi +#+TITLE: On the computation of pi #+LANGUAGE: en #+HTML_HEAD: @@ -8,7 +8,7 @@ #+HTML_HEAD: #+HTML_HEAD: -#+PROPERTY: header-args :eval never-export +#+PROPERTY: header-args :session :exports both * Asking the maths library My computer tells me that $\pi$ is /approximatively/ @@ -23,7 +23,7 @@ pi * Buffon's needle Applying the method of [[https://en.wikipedia.org/wiki/Buffon%2527s_needle_problem][Buffon's needle]], we get the *approximation* -#+begin_src R :results output :session *R* :exports results +#+begin_src R :results output :session *R* :exports both set.seed(42) N = 100000 x = runif(N) @@ -35,7 +35,7 @@ theta = pi/2*runif(N) : [1] 3.14327 * 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 $X\simU(0,1)$ and $Y\simU(0,1)$, then $P[X^2+Y^2 \le1]=\pi/4$ (see [[https://en.wikipedia.org/wiki/Monte_Carlo_method]["Monte Carlo method" on Wikipedia]]). The following code uses this approach: +A method that is easier to understand and does not make use of the $\sin$ function is based on the fact that if $X\simU(0,1)$ and $Y\simU(0,1)$, 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 R :results output graphics :file figure_pi_mc1.png :exports both :width 600 :height 400 :session *R* set.seed(42) @@ -43,13 +43,13 @@ N = 1000 df = data.frame(X = runif(N), Y = runif(N)) df$Accept = (df$X**2 + df$Y**2 <=1) library(ggplot2) -ggplot(df, aes(x=X, y=Y, color=Accept)) + geom_point(alpha=.2) + coord_fixed() + theme_bw() +ggplot(df, aes(x=X,y=Y,color=Accept)) + geom_point(alpha=.2) + coord_fixed() + theme_bw() #+end_src #+RESULTS: [[file:figure_pi_mc1.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: +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 R :results output :session *R* :exports both 4*mean(df$Accept) #+end_src