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2022-11-17 #+LANGUAGE: fr -# #+PROPERTY: header-args :eval never-export - #+HTML_HEAD: #+HTML_HEAD: #+HTML_HEAD: @@ -11,7 +9,9 @@ #+HTML_HEAD: #+HTML_HEAD: -* 1 En demandant à la lib maths +#+PROPERTY: header-args :session :exports both + +* En demandant à la lib maths Mon ordinateur m'indique que \pi vaut /approximativement/ #+begin_src R :results output :session *R* :exports both pi @@ -20,7 +20,7 @@ #+RESULTS: : [1] 3.141593 -* 2 En utilisant la méthode des aiguilles de Buffon +* En utilisant la méthode des aiguilles de Buffon Mais calculé avec la *méthode* des [[https://fr.wikipedia.org/wiki/Aiguille_de_Buffon][aiguilles de Buffon]], on obtiendrait comme *approximation* : #+begin_src R :results output :session *R* :exports both @@ -30,10 +30,10 @@ theta = pi/2*runif(N) 2/(mean(x+sin(theta)>1) #+end_src -* 3 Avec un argumennt "fréquentiel" de surface +* Avec un argumennt "fréquentiel" 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 \sim U(0,1) et Y \sim U(0,1) alors P[X^{2} + Y^{2} \leq 1] = \pi/4 (voir +$X \sim U(0,1) et Y \sim U(0,1)$ alors $P[X^{2} + Y^{2} \leq 1] = \pi/4$ (voir [[https://fr.wikipedia.org/wiki/M%C3%A9thode_de_Monte-Carlo#D%C3%A9termination_de_la_valeur_de_%CF%80][méthode de Monte Carlo sur Wikipedia]]). Le code suivant illustre ce fait : #+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R* @@ -41,4 +41,12 @@ set.seed(42) N = 1000 df = data.frame(X = runif(N), Y = runif(N)) df$Accept = (df$X**2 + df$Y**2 <= 1) -library +library(ggplot2) +ggplot(df, aes(x=X,y=Y,color=Accept)) + geom_point(alpha=.2) + coord_fixed() + theme_bw() +#+end_src + +Il est alors aisé d'obtenir une approximation (pas terrible) de \pi en +comptant combien de fois, en moyenne, $X^2 + Y^2$ est inférieur à 1 : +#+begin_src R :results output :session *R* :exports both +4*mean(df$Accept) +#+end_src -- 2.18.1