putv(A,method,tol)

function [Abar,U,k] = putv(A,method,tol)
%  PURPOSE: transforms a matrix A to [Abar; 0], where Abar has full row rank
%
% ---------------------------------------------------
%  USAGE: [Abar,U,k] = putv(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)
%
%         Abar                      matrix with full row rank
%         U                         an orthogonal matrix such that U'A =
%                                   [Abar;0]
%         k                         estimated rank of Abar
%
%   COMMENTS:
%       Version based on qr decomposition is faster but less 
%       accurate than version based on svd. If 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       
        [U,R,P]         = qr(A);
        k               = rank2(R,tol);
        Abar            = R(1:k,:)*P';
    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);
        Abar            = S(1:k,1:k)*V(:,1:k)';
end