Mercurial > octave-nkf
view libinterp/corefcn/lu.cc @ 19627:446c46af4b42 stable
strip trailing whitespace from most source files
* Makefile.am, NEWS, build-aux/common.mk, configure.ac,
doc/Makefile.am, doc/doxyhtml/Makefile.am,
doc/interpreter/Makefile.am, doc/interpreter/arith.txi,
doc/interpreter/audio.txi, doc/interpreter/basics.txi,
doc/interpreter/bugs.txi, doc/interpreter/container.txi,
doc/interpreter/cp-idx.txi, doc/interpreter/data.txi,
doc/interpreter/debug.txi, doc/interpreter/diagperm.txi,
doc/interpreter/diffeq.txi, doc/interpreter/doccheck/README,
doc/interpreter/doccheck/spellcheck, doc/interpreter/emacs.txi,
doc/interpreter/errors.txi, doc/interpreter/eval.txi,
doc/interpreter/expr.txi, doc/interpreter/external.txi,
doc/interpreter/fn-idx.txi, doc/interpreter/func.txi,
doc/interpreter/geometry.txi, doc/interpreter/geometryimages.m,
doc/interpreter/gpl.txi, doc/interpreter/grammar.txi,
doc/interpreter/gui.txi, doc/interpreter/image.txi,
doc/interpreter/install.txi, doc/interpreter/interp.txi,
doc/interpreter/interpimages.m, doc/interpreter/intro.txi,
doc/interpreter/io.txi, doc/interpreter/java.txi,
doc/interpreter/linalg.txi, doc/interpreter/macros.texi,
doc/interpreter/matrix.txi, doc/interpreter/munge-texi.pl,
doc/interpreter/nonlin.txi, doc/interpreter/numbers.txi,
doc/interpreter/obsolete.txi, doc/interpreter/octave-config.1,
doc/interpreter/octave.texi, doc/interpreter/oop.txi,
doc/interpreter/op-idx.txi, doc/interpreter/optim.txi,
doc/interpreter/package.txi, doc/interpreter/plot.txi,
doc/interpreter/poly.txi, doc/interpreter/preface.txi,
doc/interpreter/quad.txi, doc/interpreter/set.txi,
doc/interpreter/signal.txi, doc/interpreter/sparse.txi,
doc/interpreter/sparseimages.m, doc/interpreter/splineimages.m,
doc/interpreter/stats.txi, doc/interpreter/stmt.txi,
doc/interpreter/strings.txi, doc/interpreter/system.txi,
doc/interpreter/testfun.txi, doc/interpreter/tips.txi,
doc/interpreter/var.txi, doc/interpreter/vectorize.txi,
doc/liboctave/Makefile.am, doc/liboctave/array.texi,
doc/liboctave/bugs.texi, doc/liboctave/cp-idx.texi,
doc/liboctave/dae.texi, doc/liboctave/diffeq.texi,
doc/liboctave/error.texi, doc/liboctave/factor.texi,
doc/liboctave/fn-idx.texi, doc/liboctave/gpl.texi,
doc/liboctave/install.texi, doc/liboctave/intro.texi,
doc/liboctave/liboctave.texi, doc/liboctave/matvec.texi,
doc/liboctave/nleqn.texi, doc/liboctave/nlfunc.texi,
doc/liboctave/ode.texi, doc/liboctave/optim.texi,
doc/liboctave/preface.texi, doc/liboctave/quad.texi,
doc/liboctave/range.texi, doc/refcard/Makefile.am,
doc/refcard/refcard.tex, etc/HACKING, etc/NEWS.1, etc/NEWS.2,
etc/NEWS.3, etc/OLD-ChangeLogs/ChangeLog,
etc/OLD-ChangeLogs/doc-ChangeLog,
etc/OLD-ChangeLogs/scripts-ChangeLog,
etc/OLD-ChangeLogs/src-ChangeLog, etc/OLD-ChangeLogs/test-ChangeLog,
etc/PROJECTS, etc/README.Cygwin, etc/README.MacOS, etc/README.MinGW,
etc/README.gnuplot, etc/gdbinit, etc/icons/Makefile.am,
examples/@polynomial/end.m, examples/@polynomial/subsasgn.m,
examples/Makefile.am, examples/standalonebuiltin.cc,
libgui/Makefile.am, libgui/qterminal/libqterminal/README,
libgui/qterminal/libqterminal/unix/BlockArray.cpp,
libgui/qterminal/libqterminal/unix/BlockArray.h,
libgui/qterminal/libqterminal/unix/Character.h,
libgui/qterminal/libqterminal/unix/CharacterColor.h,
libgui/qterminal/libqterminal/unix/Emulation.cpp,
libgui/qterminal/libqterminal/unix/Emulation.h,
libgui/qterminal/libqterminal/unix/Filter.cpp,
libgui/qterminal/libqterminal/unix/Filter.h,
libgui/qterminal/libqterminal/unix/History.cpp,
libgui/qterminal/libqterminal/unix/History.h,
libgui/qterminal/libqterminal/unix/KeyboardTranslator.cpp,
libgui/qterminal/libqterminal/unix/KeyboardTranslator.h,
libgui/qterminal/libqterminal/unix/LineFont.h,
libgui/qterminal/libqterminal/unix/QUnixTerminalImpl.cpp,
libgui/qterminal/libqterminal/unix/QUnixTerminalImpl.h,
libgui/qterminal/libqterminal/unix/Screen.cpp,
libgui/qterminal/libqterminal/unix/Screen.h,
libgui/qterminal/libqterminal/unix/ScreenWindow.cpp,
libgui/qterminal/libqterminal/unix/ScreenWindow.h,
libgui/qterminal/libqterminal/unix/TerminalCharacterDecoder.cpp,
libgui/qterminal/libqterminal/unix/TerminalCharacterDecoder.h,
libgui/qterminal/libqterminal/unix/Vt102Emulation.h,
libgui/qterminal/libqterminal/win32/QWinTerminalImpl.cpp,
libgui/qterminal/qterminal/main.cpp,
libgui/src/m-editor/file-editor-tab.cc, libgui/src/octave-gui.cc,
libgui/src/octave-qt-link.cc, libinterp/corefcn/data.cc,
libinterp/corefcn/defun-int.h, libinterp/corefcn/det.cc,
libinterp/corefcn/gl2ps-renderer.cc, libinterp/corefcn/graphics.cc,
libinterp/corefcn/graphics.in.h, libinterp/corefcn/ls-mat5.cc,
libinterp/corefcn/lu.cc, libinterp/corefcn/oct-tex-parser.yy,
libinterp/corefcn/oct-tex-symbols.in, libinterp/corefcn/quadcc.cc,
libinterp/corefcn/zfstream.cc, libinterp/dldfcn/__eigs__.cc,
libinterp/dldfcn/__voronoi__.cc, libinterp/gendoc.pl,
libinterp/genprops.awk, libinterp/mk-errno-list, libinterp/mk-pkg-add,
libinterp/mkbuiltins, libinterp/mkdefs, libinterp/mkdocs,
libinterp/mkops, libinterp/octave-value/ov-java.cc,
libinterp/parse-tree/lex.ll, libinterp/parse-tree/oct-parse.in.yy,
libinterp/parse-tree/octave.gperf, liboctave/Makefile.am,
liboctave/array/Array.cc, liboctave/array/module.mk,
liboctave/cruft/daspk/datv.f, liboctave/cruft/daspk/dcnst0.f,
liboctave/cruft/daspk/dcnstr.f, liboctave/cruft/daspk/ddasic.f,
liboctave/cruft/daspk/ddasid.f, liboctave/cruft/daspk/ddasik.f,
liboctave/cruft/daspk/ddaspk.f, liboctave/cruft/daspk/ddstp.f,
liboctave/cruft/daspk/ddwnrm.f, liboctave/cruft/daspk/dfnrmd.f,
liboctave/cruft/daspk/dfnrmk.f, liboctave/cruft/daspk/dhels.f,
liboctave/cruft/daspk/dheqr.f, liboctave/cruft/daspk/dinvwt.f,
liboctave/cruft/daspk/dlinsd.f, liboctave/cruft/daspk/dlinsk.f,
liboctave/cruft/daspk/dmatd.f, liboctave/cruft/daspk/dnedd.f,
liboctave/cruft/daspk/dnedk.f, liboctave/cruft/daspk/dnsd.f,
liboctave/cruft/daspk/dnsid.f, liboctave/cruft/daspk/dnsik.f,
liboctave/cruft/daspk/dnsk.f, liboctave/cruft/daspk/dorth.f,
liboctave/cruft/daspk/dslvd.f, liboctave/cruft/daspk/dslvk.f,
liboctave/cruft/daspk/dspigm.f, liboctave/cruft/daspk/dyypnw.f,
liboctave/cruft/dasrt/ddasrt.f, liboctave/cruft/dasrt/drchek.f,
liboctave/cruft/dassl/ddaslv.f, liboctave/cruft/dassl/ddassl.f,
liboctave/cruft/misc/blaswrap.c, liboctave/cruft/misc/module.mk,
liboctave/cruft/odepack/cfode.f, liboctave/cruft/odepack/dlsode.f,
liboctave/cruft/odepack/ewset.f, liboctave/cruft/odepack/intdy.f,
liboctave/cruft/odepack/prepj.f, liboctave/cruft/odepack/sintdy.f,
liboctave/cruft/odepack/slsode.f, liboctave/cruft/odepack/solsy.f,
liboctave/cruft/odepack/ssolsy.f, liboctave/cruft/odepack/stode.f,
liboctave/cruft/odepack/vnorm.f, liboctave/cruft/ranlib/Basegen.doc,
liboctave/cruft/ranlib/README, liboctave/cruft/ranlib/genbet.f,
liboctave/cruft/ranlib/genexp.f, liboctave/cruft/ranlib/gennch.f,
liboctave/cruft/ranlib/gennf.f, liboctave/cruft/ranlib/gennor.f,
liboctave/cruft/ranlib/getsd.f, liboctave/cruft/ranlib/initgn.f,
liboctave/cruft/ranlib/phrtsd.f, liboctave/cruft/ranlib/randlib.fdoc,
liboctave/cruft/ranlib/setsd.f, liboctave/cruft/ranlib/tstgmn.for,
liboctave/cruft/ranlib/tstmid.for, liboctave/cruft/slatec-fn/atanh.f,
liboctave/cruft/slatec-fn/datanh.f,
liboctave/cruft/slatec-fn/xgmainc.f,
liboctave/cruft/slatec-fn/xsgmainc.f, liboctave/numeric/module.mk,
liboctave/operators/mk-ops.awk, liboctave/operators/mx-ops,
liboctave/operators/sparse-mk-ops.awk,
liboctave/operators/sparse-mx-ops, liboctave/operators/vx-ops,
liboctave/util/module.mk, run-octave.in, scripts/@ftp/ftp.m,
scripts/audio/wavread.m, scripts/deprecated/java_convert_matrix.m,
scripts/deprecated/java_debug.m, scripts/deprecated/java_invoke.m,
scripts/deprecated/java_new.m,
scripts/deprecated/java_unsigned_conversion.m,
scripts/deprecated/javafields.m, scripts/deprecated/javamethods.m,
scripts/deprecated/shell_cmd.m, scripts/general/accumarray.m,
scripts/general/display.m, scripts/general/fieldnames.m,
scripts/general/interp1.m, scripts/general/interp2.m,
scripts/general/interp3.m, scripts/general/isa.m,
scripts/general/methods.m, scripts/general/sortrows.m,
scripts/geometry/convhull.m, scripts/geometry/delaunay.m,
scripts/geometry/delaunay3.m, scripts/geometry/delaunayn.m,
scripts/geometry/griddata.m, scripts/geometry/griddatan.m,
scripts/geometry/voronoi.m, scripts/geometry/voronoin.m,
scripts/gui/guihandles.m, scripts/gui/inputdlg.m,
scripts/gui/listdlg.m, scripts/gui/msgbox.m, scripts/gui/questdlg.m,
scripts/gui/uigetfile.m, scripts/gui/waitbar.m, scripts/gui/warndlg.m,
scripts/help/doc.m, scripts/help/help.m, scripts/help/type.m,
scripts/image/bone.m, scripts/image/cmpermute.m,
scripts/image/cmunique.m, scripts/image/colorcube.m,
scripts/image/colormap.m, scripts/image/contrast.m,
scripts/image/gray2ind.m, scripts/image/image.m,
scripts/image/imshow.m, scripts/image/ind2gray.m, scripts/image/jet.m,
scripts/image/rgb2ntsc.m, scripts/image/spinmap.m,
scripts/io/importdata.m, scripts/io/strread.m, scripts/io/textread.m,
scripts/io/textscan.m, scripts/java/java_get.m,
scripts/java/java_set.m, scripts/java/javaaddpath.m,
scripts/java/javaclasspath.m, scripts/java/javamem.m,
scripts/linear-algebra/linsolve.m, scripts/linear-algebra/qzhess.m,
scripts/miscellaneous/debug.m, scripts/miscellaneous/desktop.m,
scripts/miscellaneous/dir.m, scripts/miscellaneous/dos.m,
scripts/miscellaneous/edit.m, scripts/miscellaneous/fact.m,
scripts/miscellaneous/getappdata.m, scripts/miscellaneous/inputname.m,
scripts/miscellaneous/license.m, scripts/miscellaneous/ls_command.m,
scripts/miscellaneous/run.m, scripts/miscellaneous/setfield.m,
scripts/miscellaneous/unix.m, scripts/miscellaneous/ver.m,
scripts/mk-pkg-add, scripts/mkdoc.pl,
scripts/optimization/fminsearch.m, scripts/optimization/optimset.m,
scripts/optimization/sqp.m, scripts/pkg/pkg.m,
scripts/pkg/private/create_pkgadddel.m,
scripts/pkg/private/fix_depends.m, scripts/pkg/private/install.m,
scripts/plot/appearance/axis.m, scripts/plot/appearance/box.m,
scripts/plot/appearance/clabel.m, scripts/plot/appearance/daspect.m,
scripts/plot/appearance/datetick.m, scripts/plot/appearance/grid.m,
scripts/plot/appearance/legend.m, scripts/plot/appearance/orient.m,
scripts/plot/appearance/shading.m, scripts/plot/appearance/text.m,
scripts/plot/appearance/title.m, scripts/plot/appearance/xlabel.m,
scripts/plot/appearance/ylabel.m, scripts/plot/appearance/zlabel.m,
scripts/plot/draw/area.m, scripts/plot/draw/bar.m,
scripts/plot/draw/barh.m, scripts/plot/draw/colorbar.m,
scripts/plot/draw/contour.m, scripts/plot/draw/contour3.m,
scripts/plot/draw/contourf.m, scripts/plot/draw/ellipsoid.m,
scripts/plot/draw/errorbar.m, scripts/plot/draw/ezcontour.m,
scripts/plot/draw/ezcontourf.m, scripts/plot/draw/ezmesh.m,
scripts/plot/draw/ezpolar.m, scripts/plot/draw/fill.m,
scripts/plot/draw/fplot.m, scripts/plot/draw/hist.m,
scripts/plot/draw/meshc.m, scripts/plot/draw/meshz.m,
scripts/plot/draw/pareto.m, scripts/plot/draw/patch.m,
scripts/plot/draw/peaks.m, scripts/plot/draw/pie.m,
scripts/plot/draw/pie3.m, scripts/plot/draw/plot.m,
scripts/plot/draw/plotyy.m, scripts/plot/draw/private/__bar__.m,
scripts/plot/draw/private/__contour__.m,
scripts/plot/draw/private/__errplot__.m,
scripts/plot/draw/private/__ezplot__.m,
scripts/plot/draw/private/__patch__.m,
scripts/plot/draw/private/__stem__.m, scripts/plot/draw/rectangle.m,
scripts/plot/draw/ribbon.m, scripts/plot/draw/rose.m,
scripts/plot/draw/scatter.m, scripts/plot/draw/scatter3.m,
scripts/plot/draw/semilogx.m, scripts/plot/draw/shrinkfaces.m,
scripts/plot/draw/sombrero.m, scripts/plot/draw/sphere.m,
scripts/plot/draw/stairs.m, scripts/plot/draw/stem.m,
scripts/plot/draw/stemleaf.m, scripts/plot/draw/surf.m,
scripts/plot/draw/surface.m, scripts/plot/draw/surfc.m,
scripts/plot/draw/surfl.m, scripts/plot/draw/surfnorm.m,
scripts/plot/draw/tetramesh.m, scripts/plot/draw/trimesh.m,
scripts/plot/draw/triplot.m, scripts/plot/draw/trisurf.m,
scripts/plot/util/__gnuplot_drawnow__.m,
scripts/plot/util/__plt_get_axis_arg__.m, scripts/plot/util/axes.m,
scripts/plot/util/clf.m, scripts/plot/util/copyobj.m,
scripts/plot/util/figure.m, scripts/plot/util/gcbo.m,
scripts/plot/util/graphics_toolkit.m, scripts/plot/util/hggroup.m,
scripts/plot/util/meshgrid.m, scripts/plot/util/newplot.m,
scripts/plot/util/print.m,
scripts/plot/util/private/__add_default_menu__.m,
scripts/plot/util/private/__fltk_print__.m,
scripts/plot/util/private/__gnuplot_print__.m,
scripts/plot/util/private/__print_parse_opts__.m,
scripts/plot/util/refreshdata.m, scripts/plot/util/subplot.m,
scripts/polynomial/conv.m, scripts/polynomial/poly.m,
scripts/polynomial/polyeig.m, scripts/polynomial/polyfit.m,
scripts/polynomial/polyval.m,
scripts/polynomial/private/__splinefit__.m,
scripts/polynomial/spline.m, scripts/prefs/prefdir.m,
scripts/prefs/preferences.m, scripts/prefs/private/prefsfile.m,
scripts/prefs/rmpref.m, scripts/signal/freqz.m,
scripts/signal/module.mk, scripts/sparse/eigs.m, scripts/sparse/pcg.m,
scripts/sparse/private/__sprand_impl__.m, scripts/sparse/sprand.m,
scripts/sparse/sprandn.m, scripts/sparse/spy.m, scripts/sparse/svds.m,
scripts/specfun/expint.m, scripts/specfun/factor.m,
scripts/special-matrix/gallery.m, scripts/special-matrix/hankel.m,
scripts/special-matrix/toeplitz.m, scripts/startup/inputrc,
scripts/statistics/base/kurtosis.m, scripts/statistics/base/moment.m,
scripts/statistics/base/qqplot.m, scripts/statistics/base/var.m,
scripts/statistics/distributions/betarnd.m,
scripts/statistics/distributions/binoinv.m,
scripts/statistics/distributions/binopdf.m,
scripts/statistics/distributions/binornd.m,
scripts/statistics/distributions/cauchy_rnd.m,
scripts/statistics/distributions/chi2rnd.m,
scripts/statistics/distributions/discrete_pdf.m,
scripts/statistics/distributions/discrete_rnd.m,
scripts/statistics/distributions/empirical_rnd.m,
scripts/statistics/distributions/exprnd.m,
scripts/statistics/distributions/frnd.m,
scripts/statistics/distributions/gamrnd.m,
scripts/statistics/distributions/geornd.m,
scripts/statistics/distributions/hygernd.m,
scripts/statistics/distributions/kolmogorov_smirnov_cdf.m,
scripts/statistics/distributions/laplace_cdf.m,
scripts/statistics/distributions/laplace_pdf.m,
scripts/statistics/distributions/logistic_cdf.m,
scripts/statistics/distributions/logistic_pdf.m,
scripts/statistics/distributions/lognrnd.m,
scripts/statistics/distributions/nbincdf.m,
scripts/statistics/distributions/nbininv.m,
scripts/statistics/distributions/nbinpdf.m,
scripts/statistics/distributions/nbinrnd.m,
scripts/statistics/distributions/normrnd.m,
scripts/statistics/distributions/poissinv.m,
scripts/statistics/distributions/poissrnd.m,
scripts/statistics/distributions/tinv.m,
scripts/statistics/distributions/trnd.m,
scripts/statistics/distributions/unidcdf.m,
scripts/statistics/distributions/unidpdf.m,
scripts/statistics/distributions/unidrnd.m,
scripts/statistics/distributions/unifrnd.m,
scripts/statistics/distributions/wblrnd.m,
scripts/statistics/models/module.mk,
scripts/statistics/tests/kruskal_wallis_test.m,
scripts/strings/base2dec.m, scripts/strings/deblank.m,
scripts/strings/dec2base.m, scripts/strings/dec2bin.m,
scripts/strings/dec2hex.m, scripts/strings/mat2str.m,
scripts/strings/ostrsplit.m, scripts/strings/regexptranslate.m,
scripts/strings/str2num.m, scripts/strings/strcat.m,
scripts/strings/strjoin.m, scripts/strings/strsplit.m,
scripts/strings/strtok.m, scripts/strings/strtrim.m,
scripts/strings/strtrunc.m, scripts/strings/substr.m,
scripts/testfun/__run_test_suite__.m, scripts/testfun/speed.m,
scripts/testfun/test.m, scripts/time/asctime.m,
scripts/time/datenum.m, scripts/time/datevec.m,
scripts/time/weekday.m, src/Makefile.am, test/Makefile.am,
test/build-bc-overload-tests.sh, test/build-sparse-tests.sh,
test/jit.tst, test/line-continue.tst: Strip trailing whitespace.
author | John W. Eaton <jwe@octave.org> |
---|---|
date | Tue, 20 Jan 2015 08:26:57 -0500 |
parents | 4c2465444a96 |
children | 0e1f5a750d00 |
line wrap: on
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/* Copyright (C) 1996-2013 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 "CmplxLU.h" #include "dbleLU.h" #include "fCmplxLU.h" #include "floatLU.h" #include "SparseCmplxLU.h" #include "SparsedbleLU.h" #include "defun.h" #include "error.h" #include "gripes.h" #include "oct-obj.h" #include "utils.h" #include "ov-re-sparse.h" #include "ov-cx-sparse.h" template <class MT> static octave_value get_lu_l (const base_lu<MT>& fact) { MT L = fact.L (); if (L.is_square ()) return octave_value (L, MatrixType (MatrixType::Lower)); else return L; } template <class MT> static octave_value get_lu_u (const base_lu<MT>& fact) { MT U = fact.U (); if (U.is_square () && fact.regular ()) return octave_value (U, MatrixType (MatrixType::Upper)); else return U; } DEFUN (lu, args, nargout, "-*- texinfo -*-\n\ @deftypefn {Built-in Function} {[@var{L}, @var{U}] =} lu (@var{A})\n\ @deftypefnx {Built-in Function} {[@var{L}, @var{U}, @var{P}] =} lu (@var{A})\n\ @deftypefnx {Built-in Function} {[@var{L}, @var{U}, @var{P}, @var{Q}] =} lu (@var{S})\n\ @deftypefnx {Built-in Function} {[@var{L}, @var{U}, @var{P}, @var{Q}, @var{R}] =} lu (@var{S})\n\ @deftypefnx {Built-in Function} {[@dots{}] =} lu (@var{S}, @var{thres})\n\ @deftypefnx {Built-in Function} {@var{y} =} lu (@dots{})\n\ @deftypefnx {Built-in Function} {[@dots{}] =} lu (@dots{}, \"vector\")\n\ @cindex LU decomposition\n\ Compute the LU@tie{}decomposition of @var{A}. If @var{A} is full\n\ subroutines from\n\ @sc{lapack} are used and if @var{A} is sparse then @sc{umfpack} is used. The\n\ result is returned in a permuted form, according to the optional return\n\ value @var{P}. For example, given the matrix @code{a = [1, 2; 3, 4]},\n\ \n\ @example\n\ [l, u, p] = lu (@var{a})\n\ @end example\n\ \n\ @noindent\n\ returns\n\ \n\ @example\n\ @group\n\ l =\n\ \n\ 1.00000 0.00000\n\ 0.33333 1.00000\n\ \n\ u =\n\ \n\ 3.00000 4.00000\n\ 0.00000 0.66667\n\ \n\ p =\n\ \n\ 0 1\n\ 1 0\n\ @end group\n\ @end example\n\ \n\ The matrix is not required to be square.\n\ \n\ When called with two or three output arguments and a spare input matrix,\n\ @code{lu} does not attempt to perform sparsity preserving column\n\ permutations. Called with a fourth output argument, the sparsity\n\ preserving column transformation @var{Q} is returned, such that\n\ @code{@var{P} * @var{A} * @var{Q} = @var{L} * @var{U}}.\n\ \n\ Called with a fifth output argument and a sparse input matrix,\n\ @code{lu} attempts to use a scaling factor @var{R} on the input matrix\n\ such that\n\ @code{@var{P} * (@var{R} \\ @var{A}) * @var{Q} = @var{L} * @var{U}}.\n\ This typically leads to a sparser and more stable factorization.\n\ \n\ An additional input argument @var{thres}, that defines the pivoting\n\ threshold can be given. @var{thres} can be a scalar, in which case\n\ it defines the @sc{umfpack} pivoting tolerance for both symmetric and\n\ unsymmetric cases. If @var{thres} is a 2-element vector, then the first\n\ element defines the pivoting tolerance for the unsymmetric @sc{umfpack}\n\ pivoting strategy and the second for the symmetric strategy. By default,\n\ the values defined by @code{spparms} are used ([0.1, 0.001]).\n\ \n\ Given the string argument @qcode{\"vector\"}, @code{lu} returns the values\n\ of @var{P} and @var{Q} as vector values, such that for full matrix,\n\ @code{@var{A} (@var{P},:) = @var{L} * @var{U}}, and @code{@var{R}(@var{P},:)\n\ * @var{A} (:, @var{Q}) = @var{L} * @var{U}}.\n\ \n\ With two output arguments, returns the permuted forms of the upper and\n\ lower triangular matrices, such that @code{@var{A} = @var{L} * @var{U}}.\n\ With one output argument @var{y}, then the matrix returned by the @sc{lapack}\n\ routines is returned. If the input matrix is sparse then the matrix @var{L}\n\ is embedded into @var{U} to give a return value similar to the full case.\n\ For both full and sparse matrices, @code{lu} loses the permutation\n\ information.\n\ @seealso{luupdate, chol, hess, qr, qz, schur, svd}\n\ @end deftypefn") { octave_value_list retval; int nargin = args.length (); bool issparse = (nargin > 0 && args(0).is_sparse_type ()); bool scale = (nargout == 5); if (nargin < 1 || (issparse && (nargin > 3 || nargout > 5)) || (!issparse && (nargin > 2 || nargout > 3))) { print_usage (); return retval; } bool vecout = false; Matrix thres; int n = 1; while (n < nargin && ! error_state) { if (args (n).is_string ()) { std::string tmp = args(n++).string_value (); if (! error_state ) { if (tmp.compare ("vector") == 0) vecout = true; else error ("lu: unrecognized string argument"); } } else { Matrix tmp = args(n++).matrix_value (); if (! error_state ) { if (!issparse) error ("lu: can not define pivoting threshold THRES for full matrices"); else if (tmp.nelem () == 1) { thres.resize (1,2); thres(0) = tmp(0); thres(1) = tmp(0); } else if (tmp.nelem () == 2) thres = tmp; else error ("lu: expecting 2-element vector for THRES"); } } } octave_value arg = args(0); octave_idx_type nr = arg.rows (); octave_idx_type nc = arg.columns (); int arg_is_empty = empty_arg ("lu", nr, nc); if (issparse) { if (arg_is_empty < 0) return retval; else if (arg_is_empty > 0) return octave_value_list (5, SparseMatrix ()); if (arg.is_real_type ()) { SparseMatrix m = arg.sparse_matrix_value (); if ( nargout < 4 ) { ColumnVector Qinit; Qinit.resize (nc); for (octave_idx_type i = 0; i < nc; i++) Qinit (i) = i; SparseLU fact (m, Qinit, thres, false, true); if ( nargout < 2 ) retval(0) = fact.Y (); else { retval(1) = octave_value ( fact.U () * fact.Pc_mat ().transpose (), MatrixType (MatrixType::Permuted_Upper, nc, fact.col_perm ())); PermMatrix P = fact.Pr_mat (); SparseMatrix L = fact.L (); if ( nargout < 3 ) retval(0) = octave_value ( P.transpose () * L, MatrixType (MatrixType::Permuted_Lower, nr, fact.row_perm ())); else { retval(0) = L; if ( vecout ) retval(2) = fact.Pr_vec(); else retval(2) = P; } } } else { SparseLU fact (m, thres, scale); if (scale) retval(4) = fact.R (); if (vecout) { retval(3) = fact.Pc_vec (); retval(2) = fact.Pr_vec (); } else { retval(3) = fact.Pc_mat (); retval(2) = fact.Pr_mat (); } retval(1) = octave_value (fact.U (), MatrixType (MatrixType::Upper)); retval(0) = octave_value (fact.L (), MatrixType (MatrixType::Lower)); } } else if (arg.is_complex_type ()) { SparseComplexMatrix m = arg.sparse_complex_matrix_value (); if ( nargout < 4 ) { ColumnVector Qinit; Qinit.resize (nc); for (octave_idx_type i = 0; i < nc; i++) Qinit (i) = i; SparseComplexLU fact (m, Qinit, thres, false, true); if ( nargout < 2 ) retval(0) = fact.Y (); else { retval(1) = octave_value ( fact.U () * fact.Pc_mat ().transpose (), MatrixType (MatrixType::Permuted_Upper, nc, fact.col_perm ())); PermMatrix P = fact.Pr_mat (); SparseComplexMatrix L = fact.L (); if ( nargout < 3 ) retval(0) = octave_value ( P.transpose () * L, MatrixType (MatrixType::Permuted_Lower, nr, fact.row_perm ())); else { retval(0) = L; if ( vecout ) retval(2) = fact.Pr_vec(); else retval(2) = P; } } } else { SparseComplexLU fact (m, thres, scale); if (scale) retval(4) = fact.R (); if (vecout) { retval(3) = fact.Pc_vec (); retval(2) = fact.Pr_vec (); } else { retval(3) = fact.Pc_mat (); retval(2) = fact.Pr_mat (); } retval(1) = octave_value (fact.U (), MatrixType (MatrixType::Upper)); retval(0) = octave_value (fact.L (), MatrixType (MatrixType::Lower)); } } else gripe_wrong_type_arg ("lu", arg); } else { if (arg_is_empty < 0) return retval; else if (arg_is_empty > 0) return octave_value_list (3, Matrix ()); if (arg.is_real_type ()) { if (arg.is_single_type ()) { FloatMatrix m = arg.float_matrix_value (); if (! error_state) { FloatLU fact (m); switch (nargout) { case 0: case 1: retval(0) = fact.Y (); break; case 2: { PermMatrix P = fact.P (); FloatMatrix L = P.transpose () * fact.L (); retval(1) = get_lu_u (fact); retval(0) = L; } break; case 3: default: { if (vecout) retval(2) = fact.P_vec (); else retval(2) = fact.P (); retval(1) = get_lu_u (fact); retval(0) = get_lu_l (fact); } break; } } } else { Matrix m = arg.matrix_value (); if (! error_state) { LU fact (m); switch (nargout) { case 0: case 1: retval(0) = fact.Y (); break; case 2: { PermMatrix P = fact.P (); Matrix L = P.transpose () * fact.L (); retval(1) = get_lu_u (fact); retval(0) = L; } break; case 3: default: { if (vecout) retval(2) = fact.P_vec (); else retval(2) = fact.P (); retval(1) = get_lu_u (fact); retval(0) = get_lu_l (fact); } break; } } } } else if (arg.is_complex_type ()) { if (arg.is_single_type ()) { FloatComplexMatrix m = arg.float_complex_matrix_value (); if (! error_state) { FloatComplexLU fact (m); switch (nargout) { case 0: case 1: retval(0) = fact.Y (); break; case 2: { PermMatrix P = fact.P (); FloatComplexMatrix L = P.transpose () * fact.L (); retval(1) = get_lu_u (fact); retval(0) = L; } break; case 3: default: { if (vecout) retval(2) = fact.P_vec (); else retval(2) = fact.P (); retval(1) = get_lu_u (fact); retval(0) = get_lu_l (fact); } break; } } } else { ComplexMatrix m = arg.complex_matrix_value (); if (! error_state) { ComplexLU fact (m); switch (nargout) { case 0: case 1: retval(0) = fact.Y (); break; case 2: { PermMatrix P = fact.P (); ComplexMatrix L = P.transpose () * fact.L (); retval(1) = get_lu_u (fact); retval(0) = L; } break; case 3: default: { if (vecout) retval(2) = fact.P_vec (); else retval(2) = fact.P (); retval(1) = get_lu_u (fact); retval(0) = get_lu_l (fact); } break; } } } } else gripe_wrong_type_arg ("lu", arg); } return retval; } /* %!assert(lu ([1, 2; 3, 4]), [3, 4; 1/3, 2/3], eps); %!test %! [l, u] = lu ([1, 2; 3, 4]); %! assert (l, [1/3, 1; 1, 0], sqrt (eps)); %! assert (u, [3, 4; 0, 2/3], sqrt (eps)); %!test %! [l, u, p] = lu ([1, 2; 3, 4]); %! assert (l, [1, 0; 1/3, 1], sqrt (eps)); %! assert (u, [3, 4; 0, 2/3], sqrt (eps)); %! assert (p(:,:), [0, 1; 1, 0], sqrt (eps)); %!test %! [l, u, p] = lu ([1, 2; 3, 4], "vector"); %! assert (l, [1, 0; 1/3, 1], sqrt (eps)); %! assert (u, [3, 4; 0, 2/3], sqrt (eps)); %! assert (p, [2;1], sqrt (eps)); %!test %! [l, u, p] = lu ([1, 2; 3, 4; 5, 6]); %! assert (l, [1, 0; 1/5, 1; 3/5, 1/2], sqrt (eps)); %! assert (u, [5, 6; 0, 4/5], sqrt (eps)); %! assert (p(:,:), [0, 0, 1; 1, 0, 0; 0 1 0], sqrt (eps)); %!assert (lu (single ([1, 2; 3, 4])), single ([3, 4; 1/3, 2/3]), eps ("single")) %!test %! [l, u] = lu (single ([1, 2; 3, 4])); %! assert (l, single ([1/3, 1; 1, 0]), sqrt (eps ("single"))); %! assert (u, single ([3, 4; 0, 2/3]), sqrt (eps ("single"))); %!test %! [l, u, p] = lu (single ([1, 2; 3, 4])); %! assert (l, single ([1, 0; 1/3, 1]), sqrt (eps ("single"))); %! assert (u, single ([3, 4; 0, 2/3]), sqrt (eps ("single"))); %! assert (p(:,:), single ([0, 1; 1, 0]), sqrt (eps ("single"))); %!test %! [l, u, p] = lu (single ([1, 2; 3, 4]), "vector"); %! assert (l, single ([1, 0; 1/3, 1]), sqrt (eps ("single"))); %! assert (u, single ([3, 4; 0, 2/3]), sqrt (eps ("single"))); %! assert (p, single ([2;1]), sqrt (eps ("single"))); %!test %! [l u p] = lu (single ([1, 2; 3, 4; 5, 6])); %! assert (l, single ([1, 0; 1/5, 1; 3/5, 1/2]), sqrt (eps ("single"))); %! assert (u, single ([5, 6; 0, 4/5]), sqrt (eps ("single"))); %! assert (p(:,:), single ([0, 0, 1; 1, 0, 0; 0 1 0]), sqrt (eps ("single"))); %!error lu () %!error <can not define pivoting threshold> lu ([1, 2; 3, 4], 2) %!testif HAVE_UMFPACK %! Bi = [1 2 3 4 5 2 3 6 7 8 4 5 7 8 9]; %! Bj = [1 3 4 5 6 7 8 9 11 12 13 14 15 16 17]; %! Bv = [1 1 1 1 1 1 -1 1 1 1 1 -1 1 -1 1]; %! B = sparse (Bi, Bj, Bv); %! [L, U] = lu (B); %! assert (L*U, B); %! [L, U, P] = lu(B); %! assert (P'*L*U, B); %! [L, U, P, Q] = lu (B); %! assert (P'*L*U*Q', B); */ static bool check_lu_dims (const octave_value& l, const octave_value& u, const octave_value& p) { octave_idx_type m = l.rows (), k = u.rows (), n = u.columns (); return ((l.ndims () == 2 && u.ndims () == 2 && k == l.columns ()) && k == std::min (m, n) && (p.is_undefined () || p.rows () == m)); } DEFUN (luupdate, args, , "-*- texinfo -*-\n\ @deftypefn {Built-in Function} {[@var{L}, @var{U}] =} luupdate (@var{L}, @var{U}, @var{x}, @var{y})\n\ @deftypefnx {Built-in Function} {[@var{L}, @var{U}, @var{P}] =} luupdate (@var{L}, @var{U}, @var{P}, @var{x}, @var{y})\n\ Given an LU@tie{}factorization of a real or complex matrix\n\ @w{@var{A} = @var{L}*@var{U}}, @var{L}@tie{}lower unit trapezoidal and\n\ @var{U}@tie{}upper trapezoidal, return the LU@tie{}factorization\n\ of @w{@var{A} + @var{x}*@var{y}.'}, where @var{x} and @var{y} are\n\ column vectors (rank-1 update) or matrices with equal number of columns\n\ (rank-k update).\n\ Optionally, row-pivoted updating can be used by supplying\n\ a row permutation (pivoting) matrix @var{P};\n\ in that case, an updated permutation matrix is returned.\n\ Note that if @var{L}, @var{U}, @var{P} is a pivoted LU@tie{}factorization\n\ as obtained by @code{lu}:\n\ \n\ @example\n\ [@var{L}, @var{U}, @var{P}] = lu (@var{A});\n\ @end example\n\ \n\ @noindent\n\ then a factorization of @tcode{@var{A}+@var{x}*@var{y}.'} can be obtained\n\ either as\n\ \n\ @example\n\ [@var{L1}, @var{U1}] = lu (@var{L}, @var{U}, @var{P}*@var{x}, @var{y})\n\ @end example\n\ \n\ @noindent\n\ or\n\ \n\ @example\n\ [@var{L1}, @var{U1}, @var{P1}] = lu (@var{L}, @var{U}, @var{P}, @var{x}, @var{y})\n\ @end example\n\ \n\ The first form uses the unpivoted algorithm, which is faster, but less\n\ stable. The second form uses a slower pivoted algorithm, which is more\n\ stable.\n\ \n\ The matrix case is done as a sequence of rank-1 updates;\n\ thus, for large enough k, it will be both faster and more accurate to\n\ recompute the factorization from scratch.\n\ @seealso{lu, cholupdate, qrupdate}\n\ @end deftypefn") { octave_idx_type nargin = args.length (); octave_value_list retval; bool pivoted = nargin == 5; if (nargin != 4 && nargin != 5) { print_usage (); return retval; } octave_value argl = args(0); octave_value argu = args(1); octave_value argp = pivoted ? args(2) : octave_value (); octave_value argx = args(2 + pivoted); octave_value argy = args(3 + pivoted); if (argl.is_numeric_type () && argu.is_numeric_type () && argx.is_numeric_type () && argy.is_numeric_type () && (! pivoted || argp.is_perm_matrix ())) { if (check_lu_dims (argl, argu, argp)) { PermMatrix P = (pivoted ? argp.perm_matrix_value () : PermMatrix::eye (argl.rows ())); if (argl.is_real_type () && argu.is_real_type () && argx.is_real_type () && argy.is_real_type ()) { // all real case if (argl.is_single_type () || argu.is_single_type () || argx.is_single_type () || argy.is_single_type ()) { FloatMatrix L = argl.float_matrix_value (); FloatMatrix U = argu.float_matrix_value (); FloatMatrix x = argx.float_matrix_value (); FloatMatrix y = argy.float_matrix_value (); FloatLU fact (L, U, P); if (pivoted) fact.update_piv (x, y); else fact.update (x, y); if (pivoted) retval(2) = fact.P (); retval(1) = get_lu_u (fact); retval(0) = get_lu_l (fact); } else { Matrix L = argl.matrix_value (); Matrix U = argu.matrix_value (); Matrix x = argx.matrix_value (); Matrix y = argy.matrix_value (); LU fact (L, U, P); if (pivoted) fact.update_piv (x, y); else fact.update (x, y); if (pivoted) retval(2) = fact.P (); retval(1) = get_lu_u (fact); retval(0) = get_lu_l (fact); } } else { // complex case if (argl.is_single_type () || argu.is_single_type () || argx.is_single_type () || argy.is_single_type ()) { FloatComplexMatrix L = argl.float_complex_matrix_value (); FloatComplexMatrix U = argu.float_complex_matrix_value (); FloatComplexMatrix x = argx.float_complex_matrix_value (); FloatComplexMatrix y = argy.float_complex_matrix_value (); FloatComplexLU fact (L, U, P); if (pivoted) fact.update_piv (x, y); else fact.update (x, y); if (pivoted) retval(2) = fact.P (); retval(1) = get_lu_u (fact); retval(0) = get_lu_l (fact); } else { ComplexMatrix L = argl.complex_matrix_value (); ComplexMatrix U = argu.complex_matrix_value (); ComplexMatrix x = argx.complex_matrix_value (); ComplexMatrix y = argy.complex_matrix_value (); ComplexLU fact (L, U, P); if (pivoted) fact.update_piv (x, y); else fact.update (x, y); if (pivoted) retval(2) = fact.P (); retval(1) = get_lu_u (fact); retval(0) = get_lu_l (fact); } } } else error ("luupdate: dimension mismatch"); } else error ("luupdate: L, U, X, and Y must be numeric"); return retval; } /* %!shared A, u, v, Ac, uc, vc %! A = [0.091364 0.613038 0.999083; %! 0.594638 0.425302 0.603537; %! 0.383594 0.291238 0.085574; %! 0.265712 0.268003 0.238409; %! 0.669966 0.743851 0.445057 ]; %! %! u = [0.85082; %! 0.76426; %! 0.42883; %! 0.53010; %! 0.80683 ]; %! %! v = [0.98810; %! 0.24295; %! 0.43167 ]; %! %! Ac = [0.620405 + 0.956953i 0.480013 + 0.048806i 0.402627 + 0.338171i; %! 0.589077 + 0.658457i 0.013205 + 0.279323i 0.229284 + 0.721929i; %! 0.092758 + 0.345687i 0.928679 + 0.241052i 0.764536 + 0.832406i; %! 0.912098 + 0.721024i 0.049018 + 0.269452i 0.730029 + 0.796517i; %! 0.112849 + 0.603871i 0.486352 + 0.142337i 0.355646 + 0.151496i ]; %! %! uc = [0.20351 + 0.05401i; %! 0.13141 + 0.43708i; %! 0.29808 + 0.08789i; %! 0.69821 + 0.38844i; %! 0.74871 + 0.25821i ]; %! %! vc = [0.85839 + 0.29468i; %! 0.20820 + 0.93090i; %! 0.86184 + 0.34689i ]; %! %!testif HAVE_QRUPDATE_LUU %! [L,U,P] = lu (A); %! [L,U] = luupdate (L,U,P*u,v); %! assert (norm (vec (tril (L)-L), Inf) == 0); %! assert (norm (vec (triu (U)-U), Inf) == 0); %! assert (norm (vec (P'*L*U - A - u*v.'), Inf) < norm (A)*1e1*eps); %! %!testif HAVE_QRUPDATE_LUU %! [L,U,P] = lu (Ac); %! [L,U] = luupdate (L,U,P*uc,vc); %! assert (norm (vec (tril (L)-L), Inf) == 0); %! assert (norm (vec (triu (U)-U), Inf) == 0); %! assert (norm (vec (P'*L*U - Ac - uc*vc.'), Inf) < norm (Ac)*1e1*eps); %!testif HAVE_QRUPDATE_LUU %! [L,U,P] = lu (single (A)); %! [L,U] = luupdate (L,U,P*single (u), single (v)); %! assert (norm (vec (tril (L)-L), Inf) == 0); %! assert (norm (vec (triu (U)-U), Inf) == 0); %! assert (norm (vec (P'*L*U - single (A) - single (u)*single (v).'), Inf) < norm (single (A))*1e1*eps ("single")); %! %!testif HAVE_QRUPDATE_LUU %! [L,U,P] = lu (single (Ac)); %! [L,U] = luupdate (L,U,P*single (uc),single (vc)); %! assert (norm (vec (tril (L)-L), Inf) == 0); %! assert (norm (vec (triu (U)-U), Inf) == 0); %! assert (norm (vec (P'*L*U - single (Ac) - single (uc)*single (vc).'), Inf) < norm (single (Ac))*1e1*eps ("single")); %!testif HAVE_QRUPDATE_LUU %! [L,U,P] = lu (A); %! [L,U,P] = luupdate (L,U,P,u,v); %! assert (norm (vec (tril (L)-L), Inf) == 0); %! assert (norm (vec (triu (U)-U), Inf) == 0); %! assert (norm (vec (P'*L*U - A - u*v.'), Inf) < norm (A)*1e1*eps); %! %!testif HAVE_QRUPDATE_LUU %! [L,U,P] = lu (Ac); %! [L,U,P] = luupdate (L,U,P,uc,vc); %! assert (norm (vec (tril (L)-L), Inf) == 0); %! assert (norm (vec (triu (U)-U), Inf) == 0); %! assert (norm (vec (P'*L*U - Ac - uc*vc.'), Inf) < norm (Ac)*1e1*eps); %!testif HAVE_QRUPDATE_LUU %! [L,U,P] = lu (single (A)); %! [L,U,P] = luupdate (L,U,P,single (u),single (v)); %! assert (norm (vec (tril (L)-L), Inf) == 0); %! assert (norm (vec (triu (U)-U), Inf) == 0); %! assert (norm (vec (P'*L*U - single (A) - single (u)*single (v).'), Inf) < norm (single (A))*1e1*eps ("single")); %! %!testif HAVE_QRUPDATE_LUU %! [L,U,P] = lu (single (Ac)); %! [L,U,P] = luupdate (L,U,P,single (uc),single (vc)); %! assert (norm (vec (tril (L)-L), Inf) == 0); %! assert (norm (vec (triu (U)-U), Inf) == 0); %! assert (norm (vec (P'*L*U - single (Ac) - single (uc)*single (vc).'), Inf) < norm (single (Ac))*1e1*eps ("single")); */