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////////////////////////////////////////////////////////////////////////
//
// Copyright (C) 2016-2021 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 "defun.h"
#include "error.h"
#include "lo-lapack-proto.h"
#include "ovl.h"
DEFUN (ordschur, args, ,
doc: /* -*- texinfo -*-
@deftypefn {} {[@var{UR}, @var{SR}] =} ordschur (@var{U}, @var{S}, @var{select})
Reorders the real Schur factorization (@var{U},@var{S}) obtained with the
@code{schur} function, so that selected eigenvalues appear in the upper left
diagonal blocks of the quasi triangular Schur matrix.
The logical vector @var{select} specifies the selected eigenvalues as they
appear along @var{S}'s diagonal.
For example, given the matrix @code{@var{A} = [1, 2; 3, 4]}, and its Schur
decomposition
@example
[@var{U}, @var{S}] = schur (@var{A})
@end example
@noindent
which returns
@example
@group
@var{U} =
-0.82456 -0.56577
0.56577 -0.82456
@var{S} =
-0.37228 -1.00000
0.00000 5.37228
@end group
@end example
It is possible to reorder the decomposition so that the positive eigenvalue
is in the upper left corner, by doing:
@example
[@var{U}, @var{S}] = ordschur (@var{U}, @var{S}, [0,1])
@end example
@seealso{schur, ordeig}
@end deftypefn */)
{
if (args.length () != 3)
print_usage ();
const Array<octave_idx_type> sel_arg = args(2).xoctave_idx_type_vector_value ("ordschur: SELECT must be an array of integers");
const octave_idx_type sel_n = sel_arg.numel ();
const dim_vector dimU = args(0).dims ();
const dim_vector dimS = args(1).dims ();
if (sel_n != dimU(0))
error ("ordschur: SELECT must have same length as the sides of U and S");
else if (sel_n != dimU(0) || sel_n != dimS(0) || sel_n != dimU(1)
|| sel_n != dimS(1))
error ("ordschur: U and S must be square and of equal sizes");
octave_value_list retval;
const bool double_type = args(0).is_double_type ()
|| args(1).is_double_type ();
const bool complex_type = args(0).iscomplex ()
|| args(1).iscomplex ();
#define PREPARE_ARGS(TYPE, TYPE_M, TYPE_COND) \
TYPE ## Matrix U = args(0).x ## TYPE_M ## _value \
("ordschur: U and S must be real or complex floating point matrices"); \
TYPE ## Matrix S = args(1).x ## TYPE_M ## _value \
("ordschur: U and S must be real or complex floating point matrices"); \
TYPE ## Matrix w (dim_vector (n, 1)); \
TYPE ## Matrix work (dim_vector (n, 1)); \
F77_INT m; \
F77_INT info; \
TYPE_COND cond1, cond2;
#define PREPARE_OUTPUT() \
if (info != 0) \
error ("ordschur: trsen failed"); \
\
retval = ovl (U, S);
F77_INT n = octave::to_f77_int (sel_n);
Array<F77_INT> sel (dim_vector (n, 1));
for (F77_INT i = 0; i < n; i++)
sel.xelem (i) = octave::to_f77_int (sel_arg.xelem (i));
if (double_type)
{
if (complex_type)
{
PREPARE_ARGS (Complex, complex_matrix, double)
F77_XFCN (ztrsen, ztrsen,
(F77_CONST_CHAR_ARG ("N"), F77_CONST_CHAR_ARG ("V"),
sel.data (), n, F77_DBLE_CMPLX_ARG (S.fortran_vec ()), n,
F77_DBLE_CMPLX_ARG (U.fortran_vec ()), n,
F77_DBLE_CMPLX_ARG (w.fortran_vec ()), m, cond1, cond2,
F77_DBLE_CMPLX_ARG (work.fortran_vec ()), n,
info));
PREPARE_OUTPUT()
}
else
{
PREPARE_ARGS (, matrix, double)
Matrix wi (dim_vector (n, 1));
Array<F77_INT> iwork (dim_vector (n, 1));
F77_XFCN (dtrsen, dtrsen,
(F77_CONST_CHAR_ARG ("N"), F77_CONST_CHAR_ARG ("V"),
sel.data (), n, S.fortran_vec (), n, U.fortran_vec (), n,
w.fortran_vec (), wi.fortran_vec (), m, cond1, cond2,
work.fortran_vec (), n, iwork.fortran_vec (), n, info));
PREPARE_OUTPUT ()
}
}
else
{
if (complex_type)
{
PREPARE_ARGS (FloatComplex, float_complex_matrix, float)
F77_XFCN (ctrsen, ctrsen,
(F77_CONST_CHAR_ARG ("N"), F77_CONST_CHAR_ARG ("V"),
sel.data (), n, F77_CMPLX_ARG (S.fortran_vec ()), n,
F77_CMPLX_ARG (U.fortran_vec ()), n,
F77_CMPLX_ARG (w.fortran_vec ()), m, cond1, cond2,
F77_CMPLX_ARG (work.fortran_vec ()), n,
info));
PREPARE_OUTPUT ()
}
else
{
PREPARE_ARGS (Float, float_matrix, float)
FloatMatrix wi (dim_vector (n, 1));
Array<F77_INT> iwork (dim_vector (n, 1));
F77_XFCN (strsen, strsen,
(F77_CONST_CHAR_ARG ("N"), F77_CONST_CHAR_ARG ("V"),
sel.data (), n, S.fortran_vec (), n, U.fortran_vec (), n,
w.fortran_vec (), wi.fortran_vec (), m, cond1, cond2,
work.fortran_vec (), n, iwork.fortran_vec (), n, info));
PREPARE_OUTPUT ()
}
}
#undef PREPARE_ARGS
#undef PREPARE_OUTPUT
return retval;
}
/*
%!test
%! A = [1, 2, 3, -2; 4, 5, 6, -5 ; 7, 8, 9, -5; 10, 11, 12, 4 ];
%! [U, T] = schur (A);
%! [US, TS] = ordschur (U, T, [ 0, 0, 1, 1 ]);
%! assert (US*TS*US', A, sqrt (eps));
%! assert (diag (T)(3:4), diag (TS)(1:2), sqrt (eps));
%!test
%! A = [1, 2, 3, -2; 4, 5, 6, -5 ; 7, 8, 9, -5; 10, 11, 12, 4 ];
%! [U, T] = schur (A);
%! [US, TS] = ordschur (single (U), single (T), [ 0, 0, 1, 1 ]);
%! assert (US*TS*US', A, sqrt (eps ("single")));
%! assert (diag (T)(3:4), diag (TS)(1:2), sqrt (eps ("single")));
%!test
%! A = [1, 2, 3, -2; 4, 5, 6, -5 ; 7, 8, 9, -5; 10, 11, 12, 4+3i ];
%! [U, T] = schur (A);
%! [US, TS] = ordschur (U, T, [ 0, 0, 1, 1 ]);
%! assert (US*TS*US', A, sqrt (eps));
%! assert (diag (T)(3:4), diag (TS)(1:2), sqrt (eps));
%!test
%! A = [1, 2, 3, -2; 4, 5, 6, -5 ; 7, 8, 9, -5; 10, 11, 12, 4+3i ];
%! [U, T] = schur (A);
%! [US, TS] = ordschur (single (U), single (T), [ 0, 0, 1, 1 ]);
%! assert (US*TS*US', A, sqrt (eps ("single")));
%! assert (diag (T)(3:4), diag (TS)(1:2), sqrt (eps ("single")));
*/
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