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+## 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();
+