v_dlyapsq

V_DLYAPSQ 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

0001 function v=v_dlyapsq(a,b)
0002 %V_DLYAPSQ Solves the discrete Lyapunov equation AV'VA' - V'V + BB' = 0
0003 % V is upper triangular with real non-negative diagonal entries
0004 % this is equivalent to v=chol(dlyap(a,b*b')) but better conditioned numerically
0005 
0006 %      Copyright (C) Mike Brookes 2002
0007 %      Version: $Id: v_dlyapsq.m 10865 2018-09-21 17:22:45Z dmb $
0008 %
0009 %   VOICEBOX is a MATLAB toolbox for speech processing.
0010 %   Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html
0011 %
0012 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0013 %   This program is free software; you can redistribute it and/or modify
0014 %   it under the terms of the GNU General Public License as published by
0015 %   the Free Software Foundation; either version 2 of the License, or
0016 %   (at your option) any later version.
0017 %
0018 %   This program is distributed in the hope that it will be useful,
0019 %   but WITHOUT ANY WARRANTY; without even the implied warranty of
0020 %   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
0021 %   GNU General Public License for more details.
0022 %
0023 %   You can obtain a copy of the GNU General Public License from
0024 %   http://www.gnu.org/copyleft/gpl.html or by writing to
0025 %   Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA.
0026 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0027 
0028 [q,s]=schur(a');
0029 [q,s]=rsf2csf(q,s);
0030 [qd,r]=qr(b'*q,0);
0031 % save r for testing
0032 r0=r;
0033 [m,n]=size(r);
0034 u=zeros(n,n);
0035 if m==1
0036    for i=1:n-1
0037       in=i+1:n;
0038       si=s(i,i);
0039       aa=sqrt(1-si*si');
0040       u(i,i)=r(1)/aa;
0041       u(i,in)=(u(i,i)*si'*s(i,in)+aa*r(2:end))/(eye(n-i)-si'*s(in,in));
0042       r=aa*(u(i,i)*s(i,in)+u(i,in)*s(in,in))-si*r(2:end);
0043    end
0044    u(n,n)=r/sqrt(1-s(n,n)*s(n,n)');
0045    
0046 else
0047    w=zeros(m,1); w(m)=1;
0048    em=eye(m);
0049    for i=1:n-m
0050       in=i+1:n;
0051       si=s(i,i);
0052       aa=sqrt(1-si*si');
0053       u(i,i)=r(1,1)/aa;
0054       u(i,in)=(u(i,i)*si'*s(i,in)+aa*r(1,2:end))/(eye(n-i)-si'*s(in,in));
0055       vv=aa*(u(i,i)*s(i,in)+u(i,in)*s(in,in))-si*r(1,2:end);
0056       rr=zeros(m,n-i);
0057       rr(1:m-1,:)=r(2:end,2:end);
0058       [qq,r]=qrupdate(em,rr,w,vv');
0059    end
0060    for i=n-m+1:n-1
0061       in=i+1:n;
0062       si=s(i,i);
0063       aa=sqrt(1-si*si');
0064       u(i,i)=r(1,1)/aa;
0065       u(i,in)=(u(i,i)*si'*s(i,in)+aa*r(1,2:end))/(eye(n-i)-si'*s(in,in));
0066       vv=aa*(u(i,i)*s(i,in)+u(i,in)*s(in,in))-si*r(1,2:end);
0067       rr=zeros(n-i+1,n-i);
0068       rr(1:n-i,:)=r(2:end,2:end);
0069       [qq,rr]=qrupdate(eye(n-i+1),rr,w(m-n+i:end),vv');
0070       r=rr(1:n-i,:);
0071    end
0072    
0073    u(n,n)=r/sqrt(1-s(n,n)*s(n,n)');
0074    
0075 end
0076 
0077 v=triu(qr(u*q'));
0078 dv=diag(v);
0079 ix=dv~=0;
0080 v(ix,:)=diag(abs(dv(ix))./dv(ix))*v(ix,:);
0081 if isreal(a) & isreal(b)
0082    v=real(v);
0083 end