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////////////////////////////////////////////////////////////////////////
//
// Copyright (C) 1996-2021 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/>.
//
////////////////////////////////////////////////////////////////////////
#if defined (HAVE_CONFIG_H)
# include "config.h"
#endif
#include "hess.h"
#include "defun.h"
#include "error.h"
#include "errwarn.h"
#include "ovl.h"
DEFUN (hess, args, nargout,
doc: /* -*- texinfo -*-
@deftypefn {} {@var{H} =} hess (@var{A})
@deftypefnx {} {[@var{P}, @var{H}] =} hess (@var{A})
@cindex Hessenberg decomposition
Compute the Hessenberg decomposition of the matrix @var{A}.
The Hessenberg decomposition is
@tex
$$
A = PHP^T
$$
where $P$ is a square unitary matrix ($P^TP = I$), and $H$
is upper Hessenberg ($H_{i,j} = 0, \forall i > j+1$).
@end tex
@ifnottex
@code{@var{P} * @var{H} * @var{P}' = @var{A}} where @var{P} is a square
unitary matrix (@code{@var{P}' * @var{P} = I}, using complex-conjugate
transposition) and @var{H} is upper Hessenberg
(@code{@var{H}(i, j) = 0 forall i > j+1)}.
@end ifnottex
The Hessenberg decomposition is usually used as the first step in an
eigenvalue computation, but has other applications as well
(see @nospell{Golub, Nash, and Van Loan},
IEEE Transactions on Automatic Control, 1979).
@seealso{eig, chol, lu, qr, qz, schur, svd}
@end deftypefn */)
{
if (args.length () != 1)
print_usage ();
octave_value arg = args(0);
if (arg.isempty ())
return octave_value_list (2, Matrix ());
if (arg.rows () != arg.columns ())
err_square_matrix_required ("hess", "A");
octave_value_list retval;
if (arg.is_single_type ())
{
if (arg.isreal ())
{
FloatMatrix tmp = arg.float_matrix_value ();
octave::math::hess<FloatMatrix> result (tmp);
if (nargout <= 1)
retval = ovl (result.hess_matrix ());
else
retval = ovl (result.unitary_hess_matrix (),
result.hess_matrix ());
}
else if (arg.iscomplex ())
{
FloatComplexMatrix ctmp = arg.float_complex_matrix_value ();
octave::math::hess<FloatComplexMatrix> result (ctmp);
if (nargout <= 1)
retval = ovl (result.hess_matrix ());
else
retval = ovl (result.unitary_hess_matrix (),
result.hess_matrix ());
}
}
else
{
if (arg.isreal ())
{
Matrix tmp = arg.matrix_value ();
octave::math::hess<Matrix> result (tmp);
if (nargout <= 1)
retval = ovl (result.hess_matrix ());
else
retval = ovl (result.unitary_hess_matrix (),
result.hess_matrix ());
}
else if (arg.iscomplex ())
{
ComplexMatrix ctmp = arg.complex_matrix_value ();
octave::math::hess<ComplexMatrix> result (ctmp);
if (nargout <= 1)
retval = ovl (result.hess_matrix ());
else
retval = ovl (result.unitary_hess_matrix (),
result.hess_matrix ());
}
else
err_wrong_type_arg ("hess", arg);
}
return retval;
}
/*
%!test
%! a = [1, 2, 3; 5, 4, 6; 8, 7, 9];
%! [p, h] = hess (a);
%! assert (p * h * p', a, sqrt (eps));
%!test
%! a = single ([1, 2, 3; 5, 4, 6; 8, 7, 9]);
%! [p, h] = hess (a);
%! assert (p * h * p', a, sqrt (eps ("single")));
%!error hess ()
%!error hess ([1, 2; 3, 4], 2)
%!error <must be a square matrix> hess ([1, 2; 3, 4; 5, 6])
*/
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