null2(A,method,tol)

function [N,M,k,l] = null2(A,method,tol)
%  PURPOSE: returns an orthonormal basis of the null space and range of A
%
% ---------------------------------------------------
%  USAGE: [N,M,k,l] = null2(A,method,tol)
%  where: 
%         A                         a matrix
%         method                    method used
%                                       1 - qr with pivoting
%                                       2 - svd
%         tol                       uses the tolerance tol when calculating
%                                   null subspaces (optional)
%
%         N                         an orthonormal basis of the null space
%                                   of A
%         M                         an orthonormal basis of the range of A
%         k                         null space dimension
%         l                         range dimension
%
%   COMMENTS:
%       Calculating null space using qr decomposition is facter than using
%       svd decomposition but less accurate. 
%       If one requires very precise null space
%       or null space and range are not clearly separated, then version based 
%       on svd decomposition should be used. 
%
% Copyright  (c) Pawel Kowal (2006)
% All rights reserved
% LREM_SOLVE toolbox is available free for noncommercial academic use only.
% [e-mail]

if nargin<3
    tol         = 10 * max(size(A)) * norm(A,1) * eps;
end

switch method
    case 1
        [Q,R,P]         = qr(A');
        k               = rank2(R,tol);
        N               = Q(:,k+1:end);
        M               = Q(:,1:k);
        k               = size(Q,2)-k;        
    case 2
        [U,S,V]         = svd(A);
        if size(A,1)>1 && size(A,2)>1
            s           = diag(S);
        else
            s           = S;
        end
        tol             = 10*max(size(A)') * eps(max(s));
        k               = sum(s>tol);
        N               = V(:,k+1:end);
        M               = V(:,1:k);
end
l                       = size(A,1)-k;