Mercurial > octave
view libinterp/corefcn/schur.cc @ 23219:3ac9f9ecfae5 stable
maint: Update copyright dates.
author | John W. Eaton <jwe@octave.org> |
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date | Wed, 22 Feb 2017 12:39:29 -0500 |
parents | e9a0469dedd9 |
children | 092078913d54 |
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/* Copyright (C) 1996-2017 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/>. */ #if defined (HAVE_CONFIG_H) # include "config.h" #endif #include <string> #include "schur.h" #include "defun.h" #include "error.h" #include "errwarn.h" #include "ovl.h" #include "utils.h" template <typename Matrix> static octave_value mark_upper_triangular (const Matrix& a) { octave_value retval = a; octave_idx_type n = a.rows (); assert (a.columns () == n); const typename Matrix::element_type zero = typename Matrix::element_type (); for (octave_idx_type i = 0; i < n; i++) if (a(i,i) == zero) return retval; retval.matrix_type (MatrixType::Upper); return retval; } DEFUN (schur, args, nargout, doc: /* -*- texinfo -*- @deftypefn {} {@var{S} =} schur (@var{A}) @deftypefnx {} {@var{S} =} schur (@var{A}, "real") @deftypefnx {} {@var{S} =} schur (@var{A}, "complex") @deftypefnx {} {@var{S} =} schur (@var{A}, @var{opt}) @deftypefnx {} {[@var{U}, @var{S}] =} schur (@dots{}) @cindex Schur decomposition Compute the Schur@tie{}decomposition of @var{A}. The Schur@tie{}decomposition is defined as @tex $$ S = U^T A U $$ @end tex @ifnottex @example @code{@var{S} = @var{U}' * @var{A} * @var{U}} @end example @end ifnottex where @var{U} is a unitary matrix @tex ($U^T U$ is identity) @end tex @ifnottex (@code{@var{U}'* @var{U}} is identity) @end ifnottex and @var{S} is upper triangular. The eigenvalues of @var{A} (and @var{S}) are the diagonal elements of @var{S}. If the matrix @var{A} is real, then the real Schur@tie{}decomposition is computed, in which the matrix @var{U} is orthogonal and @var{S} is block upper triangular with blocks of size at most @tex $2 \times 2$ @end tex @ifnottex @code{2 x 2} @end ifnottex along the diagonal. The diagonal elements of @var{S} (or the eigenvalues of the @tex $2 \times 2$ @end tex @ifnottex @code{2 x 2} @end ifnottex blocks, when appropriate) are the eigenvalues of @var{A} and @var{S}. The default for real matrices is a real Schur@tie{}decomposition. A complex decomposition may be forced by passing the flag @qcode{"complex"}. The eigenvalues are optionally ordered along the diagonal according to the value of @var{opt}. @code{@var{opt} = "a"} indicates that all eigenvalues with negative real parts should be moved to the leading block of @var{S} (used in @code{are}), @code{@var{opt} = "d"} indicates that all eigenvalues with magnitude less than one should be moved to the leading block of @var{S} (used in @code{dare}), and @code{@var{opt} = "u"}, the default, indicates that no ordering of eigenvalues should occur. The leading @var{k} columns of @var{U} always span the @var{A}-invariant subspace corresponding to the @var{k} leading eigenvalues of @var{S}. The Schur@tie{}decomposition is used to compute eigenvalues of a square matrix, and has applications in the solution of algebraic Riccati equations in control (see @code{are} and @code{dare}). @seealso{rsf2csf, ordschur, lu, chol, hess, qr, qz, svd} @end deftypefn */) { int nargin = args.length (); if (nargin < 1 || nargin > 2 || nargout > 2) print_usage (); octave_value arg = args(0); std::string ord; if (nargin == 2) ord = args(1).xstring_value ("schur: second argument must be a string"); bool force_complex = false; if (ord == "real") { ord = ""; } else if (ord == "complex") { force_complex = true; ord = ""; } else { char ord_char = ord.empty () ? 'U' : ord[0]; if (ord_char != 'U' && ord_char != 'A' && ord_char != 'D' && ord_char != 'u' && ord_char != 'a' && ord_char != 'd') { warning ("schur: incorrect ordered schur argument '%s'", ord.c_str ()); return ovl (); } } octave_idx_type nr = arg.rows (); octave_idx_type nc = arg.columns (); if (nr != nc) err_square_matrix_required ("schur", "A"); if (! arg.is_numeric_type ()) err_wrong_type_arg ("schur", arg); octave_value_list retval; if (arg.is_single_type ()) { if (! force_complex && arg.is_real_type ()) { FloatMatrix tmp = arg.float_matrix_value (); if (nargout <= 1) { octave::math::schur<FloatMatrix> result (tmp, ord, false); retval = ovl (result.schur_matrix ()); } else { octave::math::schur<FloatMatrix> result (tmp, ord, true); retval = ovl (result.unitary_matrix (), result.schur_matrix ()); } } else { FloatComplexMatrix ctmp = arg.float_complex_matrix_value (); if (nargout <= 1) { octave::math::schur<FloatComplexMatrix> result (ctmp, ord, false); retval = ovl (mark_upper_triangular (result.schur_matrix ())); } else { octave::math::schur<FloatComplexMatrix> result (ctmp, ord, true); retval = ovl (result.unitary_matrix (), mark_upper_triangular (result.schur_matrix ())); } } } else { if (! force_complex && arg.is_real_type ()) { Matrix tmp = arg.matrix_value (); if (nargout <= 1) { octave::math::schur<Matrix> result (tmp, ord, false); retval = ovl (result.schur_matrix ()); } else { octave::math::schur<Matrix> result (tmp, ord, true); retval = ovl (result.unitary_matrix (), result.schur_matrix ()); } } else { ComplexMatrix ctmp = arg.complex_matrix_value (); if (nargout <= 1) { octave::math::schur<ComplexMatrix> result (ctmp, ord, false); retval = ovl (mark_upper_triangular (result.schur_matrix ())); } else { octave::math::schur<ComplexMatrix> result (ctmp, ord, true); retval = ovl (result.unitary_matrix (), mark_upper_triangular (result.schur_matrix ())); } } } return retval; } /* %!test %! a = [1, 2, 3; 4, 5, 9; 7, 8, 6]; %! [u, s] = schur (a); %! assert (u' * a * u, s, sqrt (eps)); %!test %! a = single ([1, 2, 3; 4, 5, 9; 7, 8, 6]); %! [u, s] = schur (a); %! assert (u' * a * u, s, sqrt (eps ("single"))); %!error schur () %!error schur (1,2,3) %!error [a,b,c] = schur (1) %!error <must be a square matrix> schur ([1, 2, 3; 4, 5, 6]) %!error <wrong type argument 'cell'> schur ({1}) %!warning <incorrect ordered schur argument> schur ([1, 2; 3, 4], "bad_opt"); */ DEFUN (rsf2csf, args, nargout, doc: /* -*- texinfo -*- @deftypefn {} {[@var{U}, @var{T}] =} rsf2csf (@var{UR}, @var{TR}) Convert a real, upper quasi-triangular Schur@tie{}form @var{TR} to a complex, upper triangular Schur@tie{}form @var{T}. Note that the following relations hold: @tex $UR \cdot TR \cdot {UR}^T = U T U^{\dagger}$ and $U^{\dagger} U$ is the identity matrix I. @end tex @ifnottex @tcode{@var{UR} * @var{TR} * @var{UR}' = @var{U} * @var{T} * @var{U}'} and @code{@var{U}' * @var{U}} is the identity matrix I. @end ifnottex Note also that @var{U} and @var{T} are not unique. @seealso{schur} @end deftypefn */) { if (args.length () != 2 || nargout > 2) print_usage (); if (! args(0).is_numeric_type ()) err_wrong_type_arg ("rsf2csf", args(0)); if (! args(1).is_numeric_type ()) err_wrong_type_arg ("rsf2csf", args(1)); if (args(0).is_complex_type () || args(1).is_complex_type ()) error ("rsf2csf: UR and TR must be real matrices"); if (args(0).is_single_type () || args(1).is_single_type ()) { FloatMatrix u = args(0).float_matrix_value (); FloatMatrix t = args(1).float_matrix_value (); octave::math::schur<FloatComplexMatrix> cs = octave::math::rsf2csf<FloatComplexMatrix, FloatMatrix> (t, u); return ovl (cs.unitary_matrix (), cs.schur_matrix ()); } else { Matrix u = args(0).matrix_value (); Matrix t = args(1).matrix_value (); octave::math::schur<ComplexMatrix> cs = octave::math::rsf2csf<ComplexMatrix, Matrix> (t, u); return ovl (cs.unitary_matrix (), cs.schur_matrix ()); } } /* %!test %! A = [1, 1, 1, 2; 1, 2, 1, 1; 1, 1, 3, 1; -2, 1, 1, 1]; %! [u, t] = schur (A); %! [U, T] = rsf2csf (u, t); %! assert (norm (u * t * u' - U * T * U'), 0, 1e-12); %! assert (norm (A - U * T * U'), 0, 1e-12); %!test %! A = rand (10); %! [u, t] = schur (A); %! [U, T] = rsf2csf (u, t); %! assert (norm (tril (T, -1)), 0); %! assert (norm (U * U'), 1, 1e-14); %!test %! A = [0, 1;-1, 0]; %! [u, t] = schur (A); %! [U, T] = rsf2csf (u,t); %! assert (U * T * U', A, 1e-14); */