function M=RandOrthMat(n, tol)
% M = RANDORTHMAT(n)
% generates a random n x n orthogonal real matrix.
%
% M = RANDORTHMAT(n,tol)
% explicitly specifies a thresh value that measures linear dependence
% of a newly formed column with the existing columns. Defaults to 1e-6.
%
% In this version the generated matrix distribution *is* uniform over the manifold
% O(n) w.r.t. the induced R^(n^2) Lebesgue measure, at a slight computational 
% overhead (randn + normalization, as opposed to rand ). 
% 
% (c) Ofek Shilon , 2006.


    if nargin==1
	  tol=1e-6;
    end
    
    M = zeros(n); % prealloc
    
    % gram-schmidt on random column vectors
    
    vi = randn(n,1);  
    % the n-dimensional normal distribution has spherical symmetry, which implies
    % that after normalization the drawn vectors would be uniformly distributed on the
    % n-dimensional unit sphere.

    M(:,1) = vi ./ norm(vi);
    
    for i=2:n
	  nrm = 0;
	  while nrm<tol
		vi = randn(n,1);
		vi = vi -  M(:,1:i-1)  * ( M(:,1:i-1).' * vi )  ;
		nrm = norm(vi);
	  end
	  M(:,i) = vi ./ nrm;

    end %i
        
end  % RandOrthMat
    