www.pudn.com > voicebox2.zip > DLYAPSQ.M

function v=dlyapsq(a,b) 
% Solves the discrete Lyapunov equation AV'VA' - V'V +BB' =0 
% V is upper triangular with real non-negative diagonal entries 
% this is equivalent to v=chol(dlyap(a,b*b')) but better conditioned numerically 
 
%      Copyright (C) Mike Brookes 2002 
% 
%      Last modified Wed Jan 30 09:44:31 2002 
% 
%   VOICEBOX is a MATLAB toolbox for speech processing. Home page is at 
%   http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html 
% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%   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 can obtain a copy of the GNU General Public License from 
%   ftp://prep.ai.mit.edu/pub/gnu/COPYING-2.0 or by writing to 
%   Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
[q,s]=schur(a'); 
[q,s]=rsf2csf(q,s); 
[qd,r]=qr(b'*q,0); 
% save r for testing 
r0=r; 
[m,n]=size(r); 
u=zeros(n,n); 
if m==1 
   for i=1:n-1 
      in=i+1:n; 
      si=s(i,i); 
      aa=sqrt(1-si*si'); 
      u(i,i)=r(1)/aa; 
      u(i,in)=(u(i,i)*si'*s(i,in)+aa*r(2:end))/(eye(n-i)-si'*s(in,in)); 
      r=aa*(u(i,i)*s(i,in)+u(i,in)*s(in,in))-si*r(2:end); 
   end 
   u(n,n)=r/sqrt(1-s(n,n)*s(n,n)'); 
    
else 
   w=zeros(m,1); w(m)=1; 
   em=eye(m); 
   for i=1:n-m 
      in=i+1:n; 
      si=s(i,i); 
      aa=sqrt(1-si*si'); 
      u(i,i)=r(1,1)/aa; 
      u(i,in)=(u(i,i)*si'*s(i,in)+aa*r(1,2:end))/(eye(n-i)-si'*s(in,in)); 
      vv=aa*(u(i,i)*s(i,in)+u(i,in)*s(in,in))-si*r(1,2:end); 
      rr=zeros(m,n-i); 
      rr(1:m-1,:)=r(2:end,2:end); 
      [qq,r]=qrupdate(em,rr,w,vv'); 
   end 
   for i=n-m+1:n-1 
      in=i+1:n; 
      si=s(i,i); 
      aa=sqrt(1-si*si'); 
      u(i,i)=r(1,1)/aa; 
      u(i,in)=(u(i,i)*si'*s(i,in)+aa*r(1,2:end))/(eye(n-i)-si'*s(in,in)); 
      vv=aa*(u(i,i)*s(i,in)+u(i,in)*s(in,in))-si*r(1,2:end); 
      rr=zeros(n-i+1,n-i); 
      rr(1:n-i,:)=r(2:end,2:end); 
      [qq,rr]=qrupdate(eye(n-i+1),rr,w(m-n+i:end),vv'); 
      r=rr(1:n-i,:); 
   end 
    
   u(n,n)=r/sqrt(1-s(n,n)*s(n,n)'); 
    
end 
 
v=triu(qr(u*q')); 
dv=diag(v); 
ix=dv~=0; 
v(ix,:)=diag(abs(dv(ix))./dv(ix))*v(ix,:); 
if isreal(a) & isreal(b) 
   v=real(v); 
end