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########################################################################
##
## Copyright (C) 1993-2024 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## 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
## <https://www.gnu.org/licenses/>.
##
########################################################################
## -*- texinfo -*-
## @deftypefn {} {[@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 =
## Diagonal Matrix
## 1 0
## 0 1
## @end group
## @end example
##
## The Hessenberg-triangular decomposition is the first step in
## @nospell{Moler and Stewart's} QZ@tie{}decomposition algorithm.
##
## Algorithm taken from @nospell{Golub and Van Loan},
## @cite{Matrix Computations, 2nd edition}.
##
## @seealso{lu, chol, hess, qr, qz, schur, svd}
## @end deftypefn
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);
## Test input validation
%!error <Invalid call> qzhess ()
%!error <Invalid call> qzhess (1)
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