Mercurial > octave-nkf
view libinterp/corefcn/schur.cc @ 17281:bc924baa2c4e
doc: Add new @qcode macro for code samples which are quoted.
Macro handles options ("on") or properties ("position") more elegantly
than @code{"text"}.
* doc/interpreter/macros.texi: Add new @qcode macro.
* doc/interpreter/tips.txi: Add documentation about @qcode macro.
* doc/interpreter/basics.txi, doc/interpreter/container.txi,
doc/interpreter/emacs.txi, doc/interpreter/errors.txi,
doc/interpreter/eval.txi, doc/interpreter/expr.txi,
doc/interpreter/external.txi, doc/interpreter/func.txi,
doc/interpreter/grammar.txi, doc/interpreter/image.txi,
doc/interpreter/install.txi, doc/interpreter/interp.txi,
doc/interpreter/io.txi, doc/interpreter/matrix.txi,
doc/interpreter/numbers.txi, doc/interpreter/oop.txi,
doc/interpreter/package.txi, doc/interpreter/plot.txi,
doc/interpreter/quad.txi, doc/interpreter/sparse.txi,
doc/interpreter/strings.txi, doc/interpreter/system.txi,
doc/interpreter/vectorize.txi, libinterp/corefcn/balance.cc,
libinterp/corefcn/bitfcns.cc, libinterp/corefcn/cellfun.cc,
libinterp/corefcn/conv2.cc, libinterp/corefcn/data.cc,
libinterp/corefcn/debug.cc, libinterp/corefcn/defaults.cc,
libinterp/corefcn/dirfns.cc, libinterp/corefcn/dlmread.cc,
libinterp/corefcn/error.cc, libinterp/corefcn/file-io.cc,
libinterp/corefcn/find.cc, libinterp/corefcn/gammainc.cc,
libinterp/corefcn/graphics.cc, libinterp/corefcn/help.cc,
libinterp/corefcn/hex2num.cc, libinterp/corefcn/input.cc,
libinterp/corefcn/load-path.cc, libinterp/corefcn/load-save.cc,
libinterp/corefcn/ls-oct-ascii.cc, libinterp/corefcn/lu.cc,
libinterp/corefcn/luinc.cc, libinterp/corefcn/matrix_type.cc,
libinterp/corefcn/oct-hist.cc, libinterp/corefcn/pager.cc,
libinterp/corefcn/pr-output.cc, libinterp/corefcn/pt-jit.cc,
libinterp/corefcn/qz.cc, libinterp/corefcn/rand.cc,
libinterp/corefcn/regexp.cc, libinterp/corefcn/schur.cc,
libinterp/corefcn/sighandlers.cc, libinterp/corefcn/sparse.cc,
libinterp/corefcn/spparms.cc, libinterp/corefcn/str2double.cc,
libinterp/corefcn/svd.cc, libinterp/corefcn/symtab.cc,
libinterp/corefcn/syscalls.cc, libinterp/corefcn/toplev.cc,
libinterp/corefcn/tril.cc, libinterp/corefcn/typecast.cc,
libinterp/corefcn/utils.cc, libinterp/corefcn/variables.cc,
libinterp/dldfcn/__init_fltk__.cc, libinterp/dldfcn/chol.cc,
libinterp/dldfcn/colamd.cc, libinterp/dldfcn/fftw.cc, libinterp/dldfcn/qr.cc,
libinterp/dldfcn/symbfact.cc, libinterp/octave-value/ov-base.cc,
libinterp/octave-value/ov-fcn-handle.cc,
libinterp/octave-value/ov-fcn-inline.cc, libinterp/octave-value/ov-java.cc,
libinterp/octave-value/ov-range.cc, libinterp/octave-value/ov-struct.cc,
libinterp/octave-value/ov-usr-fcn.cc, libinterp/parse-tree/oct-parse.in.yy,
libinterp/parse-tree/pt-binop.cc, libinterp/parse-tree/pt-eval.cc,
libinterp/parse-tree/pt-mat.cc, scripts/@ftp/ftp.m,
scripts/deprecated/java_convert_matrix.m, scripts/deprecated/java_debug.m,
scripts/deprecated/java_unsigned_conversion.m, scripts/deprecated/shell_cmd.m,
scripts/general/dblquad.m, scripts/general/display.m,
scripts/general/genvarname.m, scripts/general/idivide.m,
scripts/general/interp1.m, scripts/general/interp2.m,
scripts/general/interp3.m, scripts/general/interpn.m, scripts/general/isa.m,
scripts/general/profexplore.m, scripts/general/profile.m,
scripts/general/quadgk.m, scripts/general/randi.m, scripts/general/structfun.m,
scripts/general/subsindex.m, scripts/general/triplequad.m,
scripts/geometry/griddata.m, scripts/geometry/griddata3.m,
scripts/geometry/griddatan.m, scripts/geometry/voronoi.m, scripts/help/help.m,
scripts/help/lookfor.m, scripts/image/cmpermute.m, scripts/image/colormap.m,
scripts/image/image.m, scripts/image/imagesc.m, scripts/image/imfinfo.m,
scripts/image/imformats.m, scripts/image/imread.m, scripts/image/imshow.m,
scripts/image/imwrite.m, scripts/image/ind2gray.m, scripts/image/lines.m,
scripts/image/rgb2ind.m, scripts/image/spinmap.m, scripts/io/dlmwrite.m,
scripts/io/strread.m, scripts/io/textread.m, scripts/io/textscan.m,
scripts/java/javaclasspath.m, scripts/java/usejava.m,
scripts/miscellaneous/bzip2.m, scripts/miscellaneous/computer.m,
scripts/miscellaneous/copyfile.m, scripts/miscellaneous/debug.m,
scripts/miscellaneous/dos.m, scripts/miscellaneous/edit.m,
scripts/miscellaneous/gzip.m, scripts/miscellaneous/license.m,
scripts/miscellaneous/mkoctfile.m, scripts/miscellaneous/movefile.m,
scripts/miscellaneous/parseparams.m, scripts/miscellaneous/unix.m,
scripts/optimization/fminbnd.m, scripts/optimization/fminsearch.m,
scripts/optimization/fminunc.m, scripts/optimization/fsolve.m,
scripts/optimization/fzero.m, scripts/optimization/glpk.m,
scripts/optimization/lsqnonneg.m, scripts/optimization/optimset.m,
scripts/optimization/pqpnonneg.m, scripts/pkg/pkg.m, scripts/plot/allchild.m,
scripts/plot/ancestor.m, scripts/plot/area.m, scripts/plot/axis.m,
scripts/plot/bar.m, scripts/plot/barh.m, scripts/plot/box.m,
scripts/plot/caxis.m, scripts/plot/cla.m, scripts/plot/clabel.m,
scripts/plot/clf.m, scripts/plot/close.m, scripts/plot/colorbar.m,
scripts/plot/daspect.m, scripts/plot/ezmesh.m, scripts/plot/ezmeshc.m,
scripts/plot/ezsurf.m, scripts/plot/ezsurfc.m, scripts/plot/findall.m,
scripts/plot/findobj.m, scripts/plot/gcbo.m, scripts/plot/gcf.m,
scripts/plot/gco.m, scripts/plot/grid.m, scripts/plot/guihandles.m,
scripts/plot/hdl2struct.m, scripts/plot/hidden.m, scripts/plot/hold.m,
scripts/plot/isonormals.m, scripts/plot/isosurface.m, scripts/plot/legend.m,
scripts/plot/mesh.m, scripts/plot/meshc.m, scripts/plot/meshz.m,
scripts/plot/newplot.m, scripts/plot/orient.m, scripts/plot/pareto.m,
scripts/plot/patch.m, scripts/plot/pbaspect.m, scripts/plot/pcolor.m,
scripts/plot/plot.m, scripts/plot/print.m,
scripts/plot/private/__add_default_menu__.m, scripts/plot/quiver.m,
scripts/plot/quiver3.m, scripts/plot/refreshdata.m, scripts/plot/saveas.m,
scripts/plot/scatter.m, scripts/plot/scatter3.m, scripts/plot/shading.m,
scripts/plot/shrinkfaces.m, scripts/plot/slice.m, scripts/plot/stem.m,
scripts/plot/stem3.m, scripts/plot/struct2hdl.m, scripts/plot/subplot.m,
scripts/plot/surf.m, scripts/plot/surfc.m, scripts/plot/surfl.m,
scripts/plot/tetramesh.m, scripts/plot/uigetfile.m, scripts/plot/uimenu.m,
scripts/plot/uiputfile.m, scripts/plot/waterfall.m, scripts/plot/whitebg.m,
scripts/plot/xlim.m, scripts/plot/ylim.m, scripts/plot/zlim.m,
scripts/polynomial/conv.m, scripts/polynomial/polyout.m,
scripts/polynomial/splinefit.m, scripts/set/ismember.m, scripts/set/powerset.m,
scripts/set/setdiff.m, scripts/set/union.m, scripts/set/unique.m,
scripts/signal/detrend.m, scripts/signal/filter2.m, scripts/signal/freqz.m,
scripts/signal/periodogram.m, scripts/signal/spectral_adf.m,
scripts/signal/spectral_xdf.m, scripts/sparse/eigs.m, scripts/sparse/svds.m,
scripts/specfun/legendre.m, scripts/special-matrix/gallery.m,
scripts/statistics/base/mean.m, scripts/statistics/base/moment.m,
scripts/statistics/tests/cor_test.m,
scripts/statistics/tests/kolmogorov_smirnov_test.m,
scripts/statistics/tests/kolmogorov_smirnov_test_2.m,
scripts/statistics/tests/kruskal_wallis_test.m,
scripts/statistics/tests/prop_test_2.m, scripts/statistics/tests/sign_test.m,
scripts/statistics/tests/t_test.m, scripts/statistics/tests/t_test_2.m,
scripts/statistics/tests/t_test_regression.m,
scripts/statistics/tests/u_test.m, scripts/statistics/tests/var_test.m,
scripts/statistics/tests/welch_test.m,
scripts/statistics/tests/wilcoxon_test.m, scripts/statistics/tests/z_test.m,
scripts/statistics/tests/z_test_2.m, scripts/strings/base2dec.m,
scripts/strings/index.m, scripts/strings/isstrprop.m,
scripts/strings/mat2str.m, scripts/strings/regexptranslate.m,
scripts/strings/rindex.m, scripts/strings/str2num.m, scripts/strings/strcat.m,
scripts/strings/strjust.m, scripts/strings/strmatch.m,
scripts/strings/validatestring.m, scripts/testfun/demo.m,
scripts/testfun/example.m, scripts/testfun/test.m, scripts/time/addtodate.m,
scripts/time/asctime.m, scripts/time/datestr.m, scripts/time/datetick.m,
scripts/time/weekday.m, scripts/ui/errordlg.m, scripts/ui/helpdlg.m,
scripts/ui/inputdlg.m, scripts/ui/listdlg.m, scripts/ui/msgbox.m,
scripts/ui/questdlg.m, scripts/ui/warndlg.m: Use new @qcode macro.
author | Rik <rik@octave.org> |
---|---|
date | Mon, 19 Aug 2013 20:46:38 -0700 |
parents | 1c21f264d26f |
children | d63878346099 |
line wrap: on
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/* Copyright (C) 1996-2012 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/>. */ #ifdef HAVE_CONFIG_H #include <config.h> #endif #include <string> #include "CmplxSCHUR.h" #include "dbleSCHUR.h" #include "fCmplxSCHUR.h" #include "floatSCHUR.h" #include "defun.h" #include "error.h" #include "gripes.h" #include "oct-obj.h" #include "utils.h" template <class 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, "-*- texinfo -*-\n\ @deftypefn {Built-in Function} {@var{S} =} schur (@var{A})\n\ @deftypefnx {Built-in Function} {@var{S} =} schur (@var{A}, \"real\")\n\ @deftypefnx {Built-in Function} {@var{S} =} schur (@var{A}, \"complex\")\n\ @deftypefnx {Built-in Function} {@var{S} =} schur (@var{A}, @var{opt})\n\ @deftypefnx {Built-in Function} {[@var{U}, @var{S}] =} schur (@var{A}, @dots{})\n\ @cindex Schur decomposition\n\ Compute the Schur@tie{}decomposition of @var{A}\n\ @tex\n\ $$\n\ S = U^T A U\n\ $$\n\ @end tex\n\ @ifnottex\n\ \n\ @example\n\ @code{@var{S} = @var{U}' * @var{A} * @var{U}}\n\ @end example\n\ \n\ @end ifnottex\n\ where @var{U} is a unitary matrix\n\ @tex\n\ ($U^T U$ is identity)\n\ @end tex\n\ @ifnottex\n\ (@code{@var{U}'* @var{U}} is identity)\n\ @end ifnottex\n\ and @var{S} is upper triangular. The eigenvalues of @var{A} (and @var{S})\n\ are the diagonal elements of @var{S}. If the matrix @var{A}\n\ is real, then the real Schur@tie{}decomposition is computed, in which the\n\ matrix @var{U} is orthogonal and @var{S} is block upper triangular\n\ with blocks of size at most\n\ @tex\n\ $2 \\times 2$\n\ @end tex\n\ @ifnottex\n\ @code{2 x 2}\n\ @end ifnottex\n\ along the diagonal. The diagonal elements of @var{S}\n\ (or the eigenvalues of the\n\ @tex\n\ $2 \\times 2$\n\ @end tex\n\ @ifnottex\n\ @code{2 x 2}\n\ @end ifnottex\n\ blocks, when appropriate) are the eigenvalues of @var{A} and @var{S}.\n\ \n\ The default for real matrices is a real Schur@tie{}decomposition.\n\ A complex decomposition may be forced by passing the flag\n\ @qcode{\"complex\"}.\n\ \n\ The eigenvalues are optionally ordered along the diagonal according to\n\ the value of @var{opt}. @code{@var{opt} = \"a\"} indicates that all\n\ eigenvalues with negative real parts should be moved to the leading\n\ block of @var{S}\n\ (used in @code{are}), @code{@var{opt} = \"d\"} indicates that all eigenvalues\n\ with magnitude less than one should be moved to the leading block of @var{S}\n\ (used in @code{dare}), and @code{@var{opt} = \"u\"}, the default, indicates\n\ that no ordering of eigenvalues should occur. The leading @var{k}\n\ columns of @var{U} always span the @var{A}-invariant\n\ subspace corresponding to the @var{k} leading eigenvalues of @var{S}.\n\ \n\ The Schur@tie{}decomposition is used to compute eigenvalues of a\n\ square matrix, and has applications in the solution of algebraic\n\ Riccati equations in control (see @code{are} and @code{dare}).\n\ @seealso{rsf2csf, lu, chol, hess, qr, qz, svd}\n\ @end deftypefn") { octave_value_list retval; int nargin = args.length (); if (nargin < 1 || nargin > 2 || nargout > 2) { print_usage (); return retval; } octave_value arg = args(0); std::string ord; if (nargin == 2) { ord = args(1).string_value (); if (error_state) { error ("schur: second argument must be a string"); return retval; } } bool force_complex = false; if (ord == "real") { ord = std::string (); } else if (ord == "complex") { force_complex = true; ord = std::string (); } 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 '%c'", ord.c_str ()); return retval; } } octave_idx_type nr = arg.rows (); octave_idx_type nc = arg.columns (); if (nr != nc) { gripe_square_matrix_required ("schur"); return retval; } if (! arg.is_numeric_type ()) gripe_wrong_type_arg ("schur", arg); else if (arg.is_single_type ()) { if (! force_complex && arg.is_real_type ()) { FloatMatrix tmp = arg.float_matrix_value (); if (! error_state) { if (nargout == 0 || nargout == 1) { FloatSCHUR result (tmp, ord, false); retval(0) = result.schur_matrix (); } else { FloatSCHUR result (tmp, ord, true); retval(1) = result.schur_matrix (); retval(0) = result.unitary_matrix (); } } } else { FloatComplexMatrix ctmp = arg.float_complex_matrix_value (); if (! error_state) { if (nargout == 0 || nargout == 1) { FloatComplexSCHUR result (ctmp, ord, false); retval(0) = mark_upper_triangular (result.schur_matrix ()); } else { FloatComplexSCHUR result (ctmp, ord, true); retval(1) = mark_upper_triangular (result.schur_matrix ()); retval(0) = result.unitary_matrix (); } } } } else { if (! force_complex && arg.is_real_type ()) { Matrix tmp = arg.matrix_value (); if (! error_state) { if (nargout == 0 || nargout == 1) { SCHUR result (tmp, ord, false); retval(0) = result.schur_matrix (); } else { SCHUR result (tmp, ord, true); retval(1) = result.schur_matrix (); retval(0) = result.unitary_matrix (); } } } else { ComplexMatrix ctmp = arg.complex_matrix_value (); if (! error_state) { if (nargout == 0 || nargout == 1) { ComplexSCHUR result (ctmp, ord, false); retval(0) = mark_upper_triangular (result.schur_matrix ()); } else { ComplexSCHUR result (ctmp, ord, true); retval(1) = mark_upper_triangular (result.schur_matrix ()); retval(0) = result.unitary_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"))); %!test %! fail ("schur ([1, 2; 3, 4], 2)", "warning"); %!error schur () %!error <argument must be a square matrix> schur ([1, 2, 3; 4, 5, 6]) */ DEFUN (rsf2csf, args, nargout, "-*- texinfo -*-\n\ @deftypefn {Function File} {[@var{U}, @var{T}] =} rsf2csf (@var{UR}, @var{TR})\n\ Convert a real, upper quasi-triangular Schur@tie{}form @var{TR} to a complex,\n\ upper triangular Schur@tie{}form @var{T}.\n\ \n\ Note that the following relations hold:\n\ \n\ @tex\n\ $UR \\cdot TR \\cdot {UR}^T = U T U^{\\dagger}$ and\n\ $U^{\\dagger} U$ is the identity matrix I.\n\ @end tex\n\ @ifnottex\n\ @tcode{@var{UR} * @var{TR} * @var{UR}' = @var{U} * @var{T} * @var{U}'} and\n\ @code{@var{U}' * @var{U}} is the identity matrix I.\n\ @end ifnottex\n\ \n\ Note also that @var{U} and @var{T} are not unique.\n\ @seealso{schur}\n\ @end deftypefn") { octave_value_list retval; if (args.length () == 2 && nargout <= 2) { if (! args(0).is_numeric_type ()) gripe_wrong_type_arg ("rsf2csf", args(0)); else if (! args(1).is_numeric_type ()) gripe_wrong_type_arg ("rsf2csf", args(1)); else if (args(0).is_complex_type () || args(1).is_complex_type ()) error ("rsf2csf: UR and TR must be real matrices"); else { 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 (); if (! error_state) { FloatComplexSCHUR cs (FloatSCHUR (t, u)); retval(1) = cs.schur_matrix (); retval(0) = cs.unitary_matrix (); } } else { Matrix u = args(0).matrix_value (); Matrix t = args(1).matrix_value (); if (! error_state) { ComplexSCHUR cs (SCHUR (t, u)); retval(1) = cs.schur_matrix (); retval(0) = cs.unitary_matrix (); } } } } else print_usage (); return retval; } /* %!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); */