%                                     tp = maxstep(dx,x,auxx,K)
% MAXSTEP  Computes maximal step length to the boundary of the cone K.
%
% SEE ALSO sedumi
% **********  INTERNAL FUNCTION OF SEDUMI **********
function tp = maxstep(dx,x,auxx,K)
 %  
 %   This file is part of SeDuMi 1.05
 %   Copyright (C) 2001 Jos F. Sturm
 %     Dept. Econometrics & O.R., Tilburg University, the Netherlands.
 %     Supported by the Netherlands Organization for Scientific Research (NWO).
 %   Affiliation SeDuMi 1.03 and 1.04Beta (2000):
 %     Dept. Quantitative Economics, Maastricht University, the Netherlands.
 %   Affiliations up to SeDuMi 1.02 (AUG1998):
 %     CRL, McMaster University, Canada.
 %     Supported by the Netherlands Organization for Scientific Research (NWO).
 % 
 %   This program is free software; you can redistribute it and/or modify
 %   it under the terms of the GNU General Public License as published by
 %   the Free Software Foundation; either version 2 of the License, or
 %   (at your option) any later version.
 % 
 %   This program is distributed in the hope that it will be useful,
 %   but WITHOUT ANY WARRANTY; without even the implied warranty of
 %   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 %   GNU General Public License for more details.
 % 
 %   You should have received a copy of the GNU General Public License
 %   along with this program; if not, write to the Free Software
 %   Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
 %
% ------------------------------------------------------------
% LP:
% ------------------------------------------------------------
 mindx = min(dx(1:K.l) ./ x(1:K.l));
% ------------------------------------------------------------
% Lorentz: compute min(eig(y)) with y:=D(x)\dx. We have
% lab1+lab2 = tr y = x'Jdx / detx,  and
% (lab2-lab1)^2 = (tr y)^2 - 4 det y = [(x'Jdx)^2 - tdetx*tdetdx] / detx^2
% ------------------------------------------------------------
 if ~isempty(K.q)
   ix = K.mainblks;
   reltr = x(ix(1):ix(2)-1).*dx(ix(1):ix(2)-1)...
       - ddot(x(ix(2):ix(3)-1),dx,K.qblkstart);
   norm2 = sqrt(reltr.^2 - tdet(dx,K).*auxx.tdet);
   mindxq = min( (reltr - norm2)./auxx.tdet);
   mindx = min(mindx, mindxq);
 end
% ------------------------------------------------------------
% PSD:
% ------------------------------------------------------------
 if ~isempty(K.s)
   reldx = psdinvscale(auxx.u, dx,K,1);
   mindxs = min(psdeig(reldx, K));
   mindx = min(mindx, mindxs);
 end
 tp = 1 / max(-mindx, 1E-16);
