[R] Re : Generate a random bistochastic matrix

[Martin Maechler]

A simpler approach --- as in similar problems ---
is to use an iterative solution  which works quite fast for any 'n'.
Interestingly, the number of necessary iterations seems to
*decrease* for increasing 'n' :

bistochMat <- function(n, tol = 1e-7, maxit = 1000)
{
    ## Purpose: Random bistochastic *square* matrix (M_{ij}):
    ##	        M_{ij} >= 0;  sum_i M_{ij} = sum_j M_{ij} = 1   (for all i, j)
    ## ----------------------------------------------------------------------
    ## Arguments: n: (n * n) matrix dimension;
    ## ----------------------------------------------------------------------
    ## Author: Martin Maechler, Date: 16 Oct 2006, 14:47
    stopifnot(maxit >= 1, tol >= 0)
    M <- matrix(runif(n*n), n,n)
    for(i in 1:maxit) {
	M <- M / outer((rM <- rowSums(M)), (cM <- colSums(M))) * sum(rM) / n
	if(all(abs(c(rM,cM) - 1) < tol))
	    break
    }
    ## cat("needed", i, "iterations\n")
    ## or rather
    attr(M, "iter") <- i
    M
}

attr(M <- bistochMat(70), "iter")
## typically:
## [1] 8

attr(M <- bistochMat(10), "iter")
## -> 13, 16, 15, 17, ...

set.seed(101) ; attr(M <- bistochMat(10), "iter")             ## 15
set.seed(101) ; attr(M <- bistochMat(10, tol = 1e-15), "iter")## 32

------------------------

Initially, I tried to solve the general [ n x m ] case.
It seems that needs a slightly smarter "bias correction"
instead of just 'sum(M) / n'.
If someone figures that out, please post your solution!

Regards,
Martin Maechler, ETH Zurich