--- /dev/null
+## Copyright (C) 1993-2012 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 3 of the License, 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, see
+## <http://www.gnu.org/licenses/>.
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {[@var{aa}, @var{bb}, @var{q}, @var{z}] =} qzhess (@var{A}, @var{B})
+## Compute the Hessenberg-triangular decomposition of the matrix pencil
+## @code{(@var{A}, @var{B})}, returning
+## @code{@var{aa} = @var{q} * @var{A} * @var{z}},
+## @code{@var{bb} = @var{q} * @var{B} * @var{z}}, with @var{q} and @var{z}
+## orthogonal. For example:
+##
+## @example
+## @group
+## [aa, bb, q, z] = qzhess ([1, 2; 3, 4], [5, 6; 7, 8])
+## @result{} aa = [ -3.02244, -4.41741; 0.92998, 0.69749 ]
+## @result{} bb = [ -8.60233, -9.99730; 0.00000, -0.23250 ]
+## @result{} q = [ -0.58124, -0.81373; -0.81373, 0.58124 ]
+## @result{} z = [ 1, 0; 0, 1 ]
+## @end group
+## @end example
+##
+## The Hessenberg-triangular decomposition is the first step in
+## Moler and Stewart's QZ@tie{}decomposition algorithm.
+##
+## Algorithm taken from Golub and Van Loan,
+## @cite{Matrix Computations, 2nd edition}.
+## @end deftypefn
+
+## 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)
+ print_usage ();
+ 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
+
+%!test
+%! a = [1 2 1 3;
+%! 2 5 3 2;
+%! 5 5 1 0;
+%! 4 0 3 2];
+%! b = [0 4 2 1;
+%! 2 3 1 1;
+%! 1 0 2 1;
+%! 2 5 3 2];
+%! mask = [0 0 0 0;
+%! 0 0 0 0;
+%! 1 0 0 0;
+%! 1 1 0 0];
+%! [aa, bb, q, z] = qzhess(a, b);
+%! assert(inv(q) - q', zeros(4), 2e-8);
+%! assert(inv(z) - z', zeros(4), 2e-8);
+%! assert(q * a * z, aa, 2e-8);
+%! assert(aa .* mask, zeros(4), 2e-8);
+%! assert(q * b * z, bb, 2e-8);
+%! assert(bb .* mask, zeros(4), 2e-8);
+
+%!test
+%! a = [1 2 3 4 5;
+%! 3 2 3 1 0;
+%! 4 3 2 1 1;
+%! 0 1 0 1 0;
+%! 3 2 1 0 5];
+%! b = [5 0 4 0 1;
+%! 1 1 1 2 5;
+%! 0 3 2 1 0;
+%! 4 3 0 3 5;
+%! 2 1 2 1 3];
+%! mask = [0 0 0 0 0;
+%! 0 0 0 0 0;
+%! 1 0 0 0 0;
+%! 1 1 0 0 0;
+%! 1 1 1 0 0];
+%! [aa, bb, q, z] = qzhess(a, b);
+%! assert(inv(q) - q', zeros(5), 2e-8);
+%! assert(inv(z) - z', zeros(5), 2e-8);
+%! assert(q * a * z, aa, 2e-8);
+%! assert(aa .* mask, zeros(5), 2e-8);
+%! assert(q * b * z, bb, 2e-8);
+%! assert(bb .* mask, zeros(5), 2e-8);
+
+%!error qzhess([0]);
+%!error qzhess();
+