`invilr` <-
function(x.ilr){
y=matrix(0,nrow=nrow(x.ilr),ncol=ncol(x.ilr)+1)
D=ncol(x.ilr)+1
y[,1]=-sqrt((D-1)/D)*x.ilr[,1]
for (i in 2:ncol(y)){
   for (j in 1:(i-1)){
      y[,i]=y[,i]+x.ilr[,j]/sqrt((D-j+1)*(D-j))
   }
}
for (i in 2:(ncol(y)-1)){
   y[,i]=y[,i]-sqrt((D-i)/(D-i+1))*x.ilr[,i]
}
yexp=exp(y)
x.back=yexp/apply(yexp,1,sum)
invisible(x.back)
#return(yexp)
}