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## Copyright (C) 1996 John W. Eaton
##
## This file is part of Octave.
##
## Octave 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, or (at your option)
## any later version.
##
## Octave 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 Octave; see the file COPYING. If not, write to the Free
## Software Foundation, 59 Temple Place - Suite 330, Boston, MA
## 02111-1307, USA.
## Usage: [aa, bb, q, z] = qzhess (a, b)
##
## Compute the qz decomposition of the matrix pencil (a - lambda b)
##
## result: (for Matlab compatibility):
##
## aa = q*a*z and bb = q*b*z, with q, z orthogonal, and
## v = matrix of generalized eigenvectors.
##
## This ought to be done in a compiled program
##
## Algorithm taken from Golub and Van Loan, Matrix Computations, 2nd ed.
## Author: A. S. Hodel <scotte@eng.auburn.edu>
## Created: August 1993
## Adapted-By: jwe
function [aa, bb, q, z] = qzhess (a, b)
if (nargin != 2)
error ("usage: [aa, bb, q, z] = qzhess (a, b)");
endif
[na, ma] = size (a);
[nb, mb] = size (b);
if (na != ma || na != nb || nb != mb)
error ("qzhess: incompatible dimensions");
endif
## Reduce to hessenberg-triangular form.
[q, bb] = qr (b);
aa = q' * a;
q = q';
z = eye (na);
for j = 1:(na-2)
for i = na:-1:(j+2)
## disp (["zero out aa(", num2str(i), ",", num2str(j), ")"])
rot = givens (aa (i-1, j), aa (i, j));
aa ((i-1):i, :) = rot *aa ((i-1):i, :);
bb ((i-1):i, :) = rot *bb ((i-1):i, :);
q ((i-1):i, :) = rot *q ((i-1):i, :);
## disp (["now zero out bb(", num2str(i), ",", num2str(i-1), ")"])
rot = givens (bb (i, i), bb (i, i-1))';
bb (:, (i-1):i) = bb (:, (i-1):i) * rot';
aa (:, (i-1):i) = aa (:, (i-1):i) * rot';
z (:, (i-1):i) = z (:, (i-1):i) * rot';
endfor
endfor
bb (2, 1) = 0.0;
for i = 3:na
bb (i, 1:(i-1)) = zeros (1, i-1);
aa (i, 1:(i-2)) = zeros (1, i-2);
endfor
endfunction
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