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Imatge:NumInt-MC.png - Viquip??dia

Imatge:NumInt-MC.png

De Viquip??dia

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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

Thomas Steiner
Permission
(Reusing this image)

Thomas Steiner put it under the GFDL


GNU head Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation license, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation license".

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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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actual17:13, 13 abr 20061.200??900 (16 KB)Thire ({{Information| |Description = numerical intergration with monte carlo. new points are dark blue (navy), old points are skyblue |Source = created with the help of GNU R, see source below |Date = 13 Apr. 2006 |Author = Thomas Steiner )

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