Image:NumInt-MC.png
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| This is a file from the Wikimedia Commons. The description on its description page there is shown below. |
| Description |
numerical intergration with monte carlo. new points are dark blue (navy), old points are skyblue. points/nodes are uniform distributed over the integration interval. the value of the integral tends to 3,32. |
|---|---|
| Source |
created with the help of GNU R, see source below |
| Date |
13 Apr. 2006 |
| Author | |
| Permission |
Thomas Steiner put it under the GFDL |
R-source code:
a=0.5
b=2
cs=c("red","skyblue","navy","darkgrey")
f<-function(x) {
return( 0.9*sin(9*x^0.6+0.3)+x+0.9 )
}
mc_plot<-function(n,new=1) {
set.seed(6911)
x=seq(a*0.6,b*1.05,length=700)
if (n>0) {
xi=sort((b-a)*runif(n)+a)
} else {
xi=NA
}
Xi=sort((b-a)*runif(new)+a)
pts=sort(c(a,xi,Xi,b))
plot(x,f(x),type="n",ylim=range(f(x),0),xlab="",ylab="")
polygon(c(pts,b,a),c(f(pts),0,0),col=cs[4], border=cs[2],lwd=2)
for (i in xi) {
segments(i,0,i,f(i),col=cs[2],lwd=3)
}
for (i in Xi) {
segments(i,0,i,f(i),col=cs[3],lwd=3)
}
lines(x,f(x),col=cs[1],lwd=3)
##calcualte area (integral)
area=0
for (p in 1:(length(pts)-1) ) {
area <- area + (pts[p+1]-pts[p]) * (f(pts[p+1])+f(pts[p])) / 2
}
text(x=0.45,y=3.2, labels=substitute(integral(f(x)*dx, A, B)%~~% AREA,list(A=a,B=b,AREA=format(area,digits=3))),cex=1.5,col=cs[4],pos=4)
}
png(filename = "NumInt-MC.png", width=1200, height=900, pointsize = 12)
par(bg="grey90",mfrow=c(2,2), oma=c(0,0,3,0), cex.axis=0.85)
mc_plot(n=0)
mc_plot(n=1)
mc_plot(n=2)
mc_plot(n=3,new=5)
title(main="Numerische Integration mit Monte Carlo (Trapezmethode)", outer=TRUE,cex.main=1.9)
dev.off()
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