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z!(7&uy$({LQUjg*9JLRdjV9OoNma9xgmq~m9^ZJP#&B*8LS%UNKYlh8nAtZS0A!dV zuy?U^)4F>R{FgrXSL4oiicf%zS^3=;e{sBRkWwqa!GB@aALe~E1QK1M9I^%`G{Z6Y z6O9<&-H7#{|K3)yhf148Pc(oqBlz}rwa0sFcI*X@> z-1qIvvjawnH@fZ90?hyC%!dS^WV^D0V=wEj`~ZIQZrt z_K{Nlbb0CKYN_pP$oU0eCDId|_E#n8y|2u_B0Ik z3lK2x0#&FPxhZn->v4U+*?y#8>*)`N@~G%&fFpXn3YY9L)!j}WjWXN*dpMz?aDBS^ z4>aeu^8Le8*1$tRKmXYrfTcrLO#kni!v9!pz=!juFYEF)XzNGKCML=FeB*Un5n6xU u>%aH2+Y2$ est inférieur à 1: +Il est alors aisé d’obtenir une approximation (pas terrible) de $\ π$ en comptant combien de fois, en moyenne, $\ X^2+Y^2$ est inférieur à 1: ```{r} 4*mean(df$Accept) diff --git a/module2/exo1/toy_document_fr.Rmd~ b/module2/exo1/toy_document_fr.Rmd~ index c15b496..196944a 100644 --- a/module2/exo1/toy_document_fr.Rmd~ +++ b/module2/exo1/toy_document_fr.Rmd~ @@ -1,7 +1,7 @@ --- -title: "Votre titre" -author: "Votre nom" -date: "La date du jour" +title: "À propos du calcul de pi" +author: "Thomas Sévère" +date: "17/03/2023" output: html_document --- @@ -11,16 +11,38 @@ knitr::opts_chunk$set(echo = TRUE) ``` -## Quelques explications +## En demandant à la lib maths +Mon ordinateur m’indique que $\ π$ vaut _approximativement_ - -```{r cars} -summary(cars) +```{r} +pi +``` +## En utilisant la méthode des aiguilles de Buffon +Mais calculé avec la **méthode** des [aiguilles de Buffon](https://fr.wikipedia.org/wiki/Aiguille_de_Buffon), on obtiendrait comme **approximation** : + +```{r} +set.seed(42) +N = 100000 +x = runif(N) +theta = pi/2*runif(N) +2/(mean(x+sin(theta)>1)) ``` -Et on peut aussi aisément inclure des figures. Par exemple: +## Avec un argument “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 $\ X∼U(0,1)$ et $\ Y∼U(0,1)$ alors $\ P[X^2+Y^2≤1]=π/4$ (voir [méthode de Monte Carlo sur Wikipedia](https://fr.wikipedia.org/wiki/M%C3%A9thode_de_Monte-Carlo#D%C3%A9termination_de_la_valeur_de_%CF%80)). Le code suivant illustre ce fait: -```{r pressure, echo=FALSE} -plot(pressure) +```{r} +set.seed(42) +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() ``` +Il est alors aisé d’obtenir une approximation (pas terrible) de $\ π$ en comptant combien de fois, en moyenne, $\ X^2+Y^2$ est inférieur à 1: + +```{r} +4*mean(df$Accept) +``` diff --git a/module2/exo1/toy_document_fr.html b/module2/exo1/toy_document_fr.html index 4c31d30..7f422aa 100644 --- a/module2/exo1/toy_document_fr.html +++ b/module2/exo1/toy_document_fr.html @@ -364,6 +364,7 @@ pre code {

En demandant à la lib maths

+

Mon ordinateur m’indique que \(\ π\) vaut approximativement

pi
## [1] 3.141593
@@ -379,7 +380,7 @@ theta = pi/2*runif(N)

Avec un argument “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∼U(0,1) et Y∼U(0,1) alors P[X2+Y2≤1]=π/4 (voir méthode de Monte Carlo sur Wikipedia). Le code suivant illustre ce fait:

+

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 \(\ X∼U(0,1)\) et \(\ Y∼U(0,1)\) alors \(\ P[X^2+Y^2≤1]=π/4\) (voir méthode de Monte Carlo sur Wikipedia). Le code suivant illustre ce fait:

set.seed(42)
 N = 1000
 df = data.frame(X = runif(N), Y = runif(N))
@@ -387,7 +388,7 @@ 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()

-

Il est alors aisé d’obtenir une approximation (pas terrible) de π en comptant combien de fois, en moyenne, \(\X<sup>2</sup>+Y<sup>2</sup>\) est inférieur à 1:

+

Il est alors aisé d’obtenir une approximation (pas terrible) de \(\ π\) en comptant combien de fois, en moyenne, \(\ X^2+Y^2\) est inférieur à 1:

4*mean(df$Accept)
## [1] 3.156
diff --git a/module2/exo1/toy_document_fr.md b/module2/exo1/toy_document_fr.md new file mode 100644 index 0000000..5bf13e3 --- /dev/null +++ b/module2/exo1/toy_document_fr.md @@ -0,0 +1,63 @@ +--- +title: "À propos du calcul de pi" +author: "Thomas Sévère" +date: "17/03/2023" +output: html_document +--- + + + + +## En demandant à la lib maths + + + +```r +pi +``` + +``` +## [1] 3.141593 +``` +## En utilisant la méthode des aiguilles de Buffon +Mais calculé avec la **méthode** des [aiguilles de Buffon](https://fr.wikipedia.org/wiki/Aiguille_de_Buffon), on obtiendrait comme **approximation** : + + +```r +set.seed(42) +N = 100000 +x = runif(N) +theta = pi/2*runif(N) +2/(mean(x+sin(theta)>1)) +``` + +``` +## [1] 3.14327 +``` + +## Avec un argument “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 $\ X∼U(0,1)$ et $\ Y∼U(0,1)$ alors $\ P[X^2+Y^2≤1]=π/4$ (voir [méthode de Monte Carlo sur Wikipedia](https://fr.wikipedia.org/wiki/M%C3%A9thode_de_Monte-Carlo#D%C3%A9termination_de_la_valeur_de_%CF%80)). Le code suivant illustre ce fait: + + +```r +set.seed(42) +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() +``` + +![plot of chunk unnamed-chunk-3](figure/unnamed-chunk-3-1.png) + +Il est alors aisé d’obtenir une approximation (pas terrible) de $\ π$ en comptant combien de fois, en moyenne, $\ X^2+Y^2$ est inférieur à 1: + + +```r +4*mean(df$Accept) +``` + +``` +## [1] 3.156 +``` -- 2.18.1