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function [A,B,Q,Z] = qzswitch(i,A,B,Q,Z)
%function [A,B,Q,Z] = qzswitch(i,A,B,Q,Z)
%
% Takes U.T. matrices A, B, orthonormal matrices Q,Z, interchanges
% diagonal elements i and i+1 of both A and B, while maintaining
% Q'AZ' and Q'BZ' unchanged. If diagonal elements of A and B
% are zero at matching positions, the returned A will have zeros at both
% positions on the diagonal. This is natural behavior if this routine is used
% to drive all zeros on the diagonal of A to the lower right, but in this case
% the qz transformation is not unique and it is not possible simply to switch
% the positions of the diagonal elements of both A and B.
%
% Copyright (C) 1997-2012 Tao Zha
%
% This 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 3 of the License, or
% (at your option) any later version.
%
% It 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.
%
% If you did not received a copy of the GNU General Public License
% with this software, see <http://www.gnu.org/licenses/>.
%
realsmall=sqrt(eps)*10;
%realsmall=1e-3;
a = A(i,i); d = B(i,i); b = A(i,i+1); e = B(i,i+1);
c = A(i+1,i+1); f = B(i+1,i+1);
% A(i:i+1,i:i+1)=[a b; 0 c];
% B(i:i+1,i:i+1)=[d e; 0 f];
if (abs(c)<realsmall & abs(f)<realsmall)
if abs(a)<realsmall
% l.r. coincident 0's with u.l. of A=0; do nothing
return
else
% l.r. coincident zeros; put 0 in u.l. of a
wz=[b; -a];
wz=wz/sqrt(wz'*wz);
wz=[wz [wz(2)';-wz(1)'] ];
xy=eye(2);
end
elseif (abs(a)<realsmall & abs(d)<realsmall)
if abs(c)<realsmall
% u.l. coincident zeros with l.r. of A=0; do nothing
return
else
% u.l. coincident zeros; put 0 in l.r. of A
wz=eye(2);
xy=[c -b];
xy=xy/sqrt(xy*xy');
xy=[[xy(2)' -xy(1)'];xy];
end
else
% usual case
wz = [c*e-f*b, (c*d-f*a)'];
xy = [(b*d-e*a)', (c*d-f*a)'];
n = sqrt(wz*wz');
m = sqrt(xy*xy');
if m<eps*100
% all elements of A and B proportional
return
end
wz = n\wz;
xy = m\xy;
wz = [wz; -wz(2)', wz(1)'];
xy = [xy;-xy(2)', xy(1)'];
end
A(i:i+1,:) = xy*A(i:i+1,:);
B(i:i+1,:) = xy*B(i:i+1,:);
A(:,i:i+1) = A(:,i:i+1)*wz;
B(:,i:i+1) = B(:,i:i+1)*wz;
Z(:,i:i+1) = Z(:,i:i+1)*wz;
Q(i:i+1,:) = xy*Q(i:i+1,:);
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