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// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
// Copyright (C) INRIA -
//
// This file must be used under the terms of the CeCILL.
// This source file is licensed as described in the file COPYING, which
// you should have received as part of this distribution. The terms
// are also available at
// http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt
function [t,siz]=equil1(p,q,tol)
//
[n,n]=size(p);
[lhs,rhs]=argn(0);
// t1
if rhs==2 then
[u,p,u,np]=svd(p);
else
[u,p,u,np]=svd(p,tol);
end;
p=sqrt(diag(p));p=[p(1:np);ones(n-np,1)];
t1=diag(ones(n,1)./p)*u';
//
// t2
q=diag(p)*(u'*q*u)*diag(p) // q1=t1**-1'*q*t1**-1
if rhs==2 then
[u,sigma1,u,nq]=svd(q(1:np,1:np))
else
[u,sigma1,u,nq]=svd(q(1:np,1:np),tol)
end;
sigma1=sqrt(diag(sigma1(1:nq,1:nq)))
t2=[u',0*ones(np,n-np);0*ones(n-np,np) eye(n-np,n-np)];
//
// t3
q=t2*q*t2';t3=eye(n,n);
if np<>n then
x=diag(ones(nq,1)./(sigma1.*sigma1))*q(1:nq,np+1:n)
t3(1:nq,np+1:n)=-x;
q=t3'*q*t3;// here t3 value is in reality t3**-1
t3(1:nq,np+1:n)=x;
end;
//
// t4
if rhs==2 then
[u,q,u,n3]=svd(q(np+1:n,np+1:n));
else
[u,q,u,n3]=svd(q(np+1:n,np+1:n),tol);
end;
t4=[diag(sqrt(sigma1)) 0*ones(nq,n-nq);
0*ones(np-nq,nq) eye(np-nq,np-nq) 0*ones(np-nq,n-np);
0*ones(n-np,np) u']
t=t4*t3*t2*t1
siz=[nq,np-nq,n3]
endfunction
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