Mercurial > octave-nkf
view scripts/sparse/eigs.m @ 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 | 1782f5294fdc |
children | 0e1f5a750d00 08dbf53f34c1 |
line wrap: on
line source
## Copyright (C) 2005-2013 David Bateman ## ## 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/>. ## -*- texinfo -*- ## @deftypefn {Function File} {@var{d} =} eigs (@var{A}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{A}, @var{k}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{A}, @var{k}, @var{sigma}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{A}, @var{k}, @var{sigma}, @var{opts}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{A}, @var{B}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{A}, @var{B}, @var{k}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{A}, @var{B}, @var{k}, @var{sigma}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{A}, @var{B}, @var{k}, @var{sigma}, @var{opts}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{af}, @var{n}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{af}, @var{n}, @var{B}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{af}, @var{n}, @var{k}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{af}, @var{n}, @var{B}, @var{k}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{af}, @var{n}, @var{k}, @var{sigma}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{af}, @var{n}, @var{B}, @var{k}, @var{sigma}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{af}, @var{n}, @var{k}, @var{sigma}, @var{opts}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{af}, @var{n}, @var{B}, @var{k}, @var{sigma}, @var{opts}) ## @deftypefnx {Function File} {[@var{V}, @var{d}] =} eigs (@var{A}, @dots{}) ## @deftypefnx {Function File} {[@var{V}, @var{d}] =} eigs (@var{af}, @var{n}, @dots{}) ## @deftypefnx {Function File} {[@var{V}, @var{d}, @var{flag}] =} eigs (@var{A}, @dots{}) ## @deftypefnx {Function File} {[@var{V}, @var{d}, @var{flag}] =} eigs (@var{af}, @var{n}, @dots{}) ## Calculate a limited number of eigenvalues and eigenvectors of @var{A}, ## based on a selection criteria. The number of eigenvalues and eigenvectors to ## calculate is given by @var{k} and defaults to 6. ## ## By default, @code{eigs} solve the equation ## @tex ## $A \nu = \lambda \nu$, ## @end tex ## @ifinfo ## @code{A * v = lambda * v}, ## @end ifinfo ## where ## @tex ## $\lambda$ is a scalar representing one of the eigenvalues, and $\nu$ ## @end tex ## @ifinfo ## @code{lambda} is a scalar representing one of the eigenvalues, and @code{v} ## @end ifinfo ## is the corresponding eigenvector. If given the positive definite matrix ## @var{B} then @code{eigs} solves the general eigenvalue equation ## @tex ## $A \nu = \lambda B \nu$. ## @end tex ## @ifinfo ## @code{A * v = lambda * B * v}. ## @end ifinfo ## ## The argument @var{sigma} determines which eigenvalues are returned. ## @var{sigma} can be either a scalar or a string. When @var{sigma} is a ## scalar, the @var{k} eigenvalues closest to @var{sigma} are returned. If ## @var{sigma} is a string, it must have one of the following values. ## ## @table @asis ## @item @qcode{"lm"} ## Largest Magnitude (default). ## ## @item @qcode{"sm"} ## Smallest Magnitude. ## ## @item @qcode{"la"} ## Largest Algebraic (valid only for real symmetric problems). ## ## @item @qcode{"sa"} ## Smallest Algebraic (valid only for real symmetric problems). ## ## @item @qcode{"be"} ## Both Ends, with one more from the high-end if @var{k} is odd (valid only for ## real symmetric problems). ## ## @item @qcode{"lr"} ## Largest Real part (valid only for complex or unsymmetric problems). ## ## @item @qcode{"sr"} ## Smallest Real part (valid only for complex or unsymmetric problems). ## ## @item @qcode{"li"} ## Largest Imaginary part (valid only for complex or unsymmetric problems). ## ## @item @qcode{"si"} ## Smallest Imaginary part (valid only for complex or unsymmetric problems). ## @end table ## ## If @var{opts} is given, it is a structure defining possible options that ## @code{eigs} should use. The fields of the @var{opts} structure are: ## ## @table @code ## @item issym ## If @var{af} is given, then flags whether the function @var{af} defines a ## symmetric problem. It is ignored if @var{A} is given. The default is false. ## ## @item isreal ## If @var{af} is given, then flags whether the function @var{af} defines a ## real problem. It is ignored if @var{A} is given. The default is true. ## ## @item tol ## Defines the required convergence tolerance, calculated as ## @code{tol * norm (A)}. The default is @code{eps}. ## ## @item maxit ## The maximum number of iterations. The default is 300. ## ## @item p ## The number of Lanzcos basis vectors to use. More vectors will result in ## faster convergence, but a greater use of memory. The optimal value of ## @code{p} is problem dependent and should be in the range @var{k} to @var{n}. ## The default value is @code{2 * @var{k}}. ## ## @item v0 ## The starting vector for the algorithm. An initial vector close to the ## final vector will speed up convergence. The default is for @sc{arpack} ## to randomly generate a starting vector. If specified, @code{v0} must be ## an @var{n}-by-1 vector where @code{@var{n} = rows (@var{A})} ## ## @item disp ## The level of diagnostic printout (0|1|2). If @code{disp} is 0 then ## diagnostics are disabled. The default value is 0. ## ## @item cholB ## Flag if @code{chol (@var{B})} is passed rather than @var{B}. The default is ## false. ## ## @item permB ## The permutation vector of the Cholesky@tie{}factorization of @var{B} if ## @code{cholB} is true. That is @code{chol (@var{B}(permB, permB))}. The ## default is @code{1:@var{n}}. ## ## @end table ## ## It is also possible to represent @var{A} by a function denoted @var{af}. ## @var{af} must be followed by a scalar argument @var{n} defining the length ## of the vector argument accepted by @var{af}. @var{af} can be ## a function handle, an inline function, or a string. When @var{af} is a ## string it holds the name of the function to use. ## ## @var{af} is a function of the form @code{y = af (x)} ## where the required return value of @var{af} is determined by ## the value of @var{sigma}. The four possible forms are ## ## @table @code ## @item A * x ## if @var{sigma} is not given or is a string other than "sm". ## ## @item A \ x ## if @var{sigma} is 0 or "sm". ## ## @item (A - sigma * I) \ x ## for the standard eigenvalue problem, where @code{I} is the identity matrix of ## the same size as @var{A}. ## ## @item (A - sigma * B) \ x ## for the general eigenvalue problem. ## @end table ## ## The return arguments of @code{eigs} depend on the number of return arguments ## requested. With a single return argument, a vector @var{d} of length @var{k} ## is returned containing the @var{k} eigenvalues that have been found. With ## two return arguments, @var{V} is a @var{n}-by-@var{k} matrix whose columns ## are the @var{k} eigenvectors corresponding to the returned eigenvalues. The ## eigenvalues themselves are returned in @var{d} in the form of a ## @var{n}-by-@var{k} matrix, where the elements on the diagonal are the ## eigenvalues. ## ## Given a third return argument @var{flag}, @code{eigs} returns the status ## of the convergence. If @var{flag} is 0 then all eigenvalues have converged. ## Any other value indicates a failure to converge. ## ## This function is based on the @sc{arpack} package, written by R. Lehoucq, ## K. Maschhoff, D. Sorensen, and C. Yang. For more information see ## @url{http://www.caam.rice.edu/software/ARPACK/}. ## ## @seealso{eig, svds} ## @end deftypefn function varargout = eigs (varargin) ## For compatibility with Matlab, handle small matrix cases here ## that ARPACK does not. if (nargin == 0) print_usage (); endif call_eig = false; offset = 0; k = 6; sigma = "lm"; if (isnumeric (varargin{1}) && issquare (varargin{1})) a = varargin{1}; if (nargin > 1 && isnumeric (varargin{2}) && issquare (varargin{2}) && size_equal (a, varargin{2})) b = varargin{2}; offset = 1; endif if (rows (a) < 9) call_eig = true; endif if (nargin > 1 + offset) tmp = varargin{2+offset}; if (isnumeric (tmp) && isscalar (tmp) && isreal (tmp) && round (tmp) == tmp) k = tmp; if (rows (a) - k < 3) call_eig = true; endif else call_eig = false; endif if (nargin > 2 + offset) tmp = varargin{3+offset}; if (ischar (tmp) || (isnumeric (tmp) && isscalar (tmp))) sigma = tmp; else call_eig = false; endif endif endif endif if (call_eig) varargout = cell (1, min (2, max (1, nargout))); if (offset) real_valued = isreal (a) && isreal (b); symmetric = issymmetric (a) && issymmetric (b); [varargout{:}] = eig (a, b); else real_valued = isreal (a); symmetric = issymmetric (a); [varargout{:}] = eig (a); endif varargout = select (varargout, k, sigma, real_valued, symmetric); if (nargout == 3) varargout{3} = 0; endif else varargout = cell (1, max (1, nargout)); [varargout{:}] = __eigs__ (varargin{:}); endif endfunction function out = select (args, k, sigma, real_valued, symmetric) if (numel (args) == 1) d = args{1}; else d = diag (args{2}); endif if (ischar (sigma)) switch (sigma) case "lm" [~, idx] = sort (abs (d), "descend"); case "sm" [~, idx] = sort (abs (d), "ascend"); case "la" if (real_valued && symmetric) [~, idx] = sort (real (d), "descend"); else error ('sigma = "la" requires real symmetric problem'); endif case "sa" if (real_valued && symmetric) [~, idx] = sort (real (d), "ascend"); else error ('sigma = "sa" requires real symmetric problem'); endif case "be" if (real_valued && symmetric) [~, idx] = sort (real (d), "ascend"); else error ('sigma = "be" requires real symmetric problem'); endif case "lr" if (! (real_valued || symmetric)) [~, idx] = sort (real (d), "descend"); else error ('sigma = "lr" requires complex or unsymmetric problem'); endif case "sr" if (! (real_valued || symmetric)) [~, idx] = sort (real (d), "ascend"); else error ('sigma = "sr" requires complex or unsymmetric problem'); endif case "li" if (! (real_valued || symmetric)) [~, idx] = sort (imag (d), "descend"); else error ('sigma = "li" requires complex or unsymmetric problem'); endif case "si" if (! (real_valued || symmetric)) [~, idx] = sort (imag (d), "ascend"); else error ('sigma = "si" requires complex or unsymmetric problem'); endif otherwise error ("unrecognized value for sigma: %s", sigma); endswitch else ## numeric sigma, find k closest values [~, idx] = sort (abs (d - sigma)); endif d = d(idx); n = numel (d); k = min (k, n); if (strcmp (sigma, "be")) tmp = k / 2; n1 = floor (tmp); n2 = n - ceil (tmp) + 1; selection = [1:floor(k/2), n2:n]; else selection = 1:k; endif d = d(selection); if (numel (args) == 1) out{1} = d; else out{2} = diag (d); v = args{1}; v = v(:,idx); out{1} = v(:,selection); endif endfunction #### SPARSE MATRIX VERSIONS #### ## Real positive definite tests, n must be even %!shared n, k, A, d0, d2 %! n = 20; %! k = 4; %! A = sparse ([3:n,1:n,1:(n-2)],[1:(n-2),1:n,3:n],[ones(1,n-2),4*ones(1,n),ones(1,n-2)]); %! d0 = eig (A); %! d2 = sort (d0); %! [~, idx] = sort (abs (d0)); %! d0 = d0(idx); %! rand ("state", 42); # initialize generator to make eigs behavior reproducible %!testif HAVE_ARPACK %! d1 = eigs (A, k); %! assert (d1, d0(end:-1:(end-k+1)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k+1); %! assert (d1, d0(end:-1:(end-k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lm"); %! assert (d1, d0(end:-1:(end-k+1)), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! d1 = eigs (A, k, "sm"); %! assert (d1, d0(k:-1:1), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "la"); %! assert (d1, d2(end:-1:(end-k+1)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "sa"); %! assert (d1, d2(1:k), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "be"); %! assert (d1, d2([1:floor(k/2), (end - ceil(k/2) + 1):end]), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k+1, "be"); %! assert (d1, d2([1:floor((k+1)/2), (end - ceil((k+1)/2) + 1):end]), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! d1 = eigs (A, k, 4.1); %! [~, idx0] = sort (abs (d0 - 4.1)); %! [~, idx1] = sort (abs (d1 - 4.1)); %! assert (d1(idx1), d0(idx0(1:k)), 1e-11); %!testif HAVE_ARPACK, HAVE_CHOLMOD %! d1 = eigs (A, speye (n), k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! assert (eigs (A, k, 4.1), eigs (A, speye (n), k, 4.1), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! d1 = eigs (A, speye (n), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, speye (n)(q,q), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! opts.cholB = true; %! d1 = eigs (A, speye (n), k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, speye (n)(q,q), k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! assert (eigs (A, k, 4.1), eigs (A, speye (n), k, 4.1), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A * x; %! opts.issym = 1; opts.isreal = 1; %! d1 = eigs (fn, n, k, "lm", opts); %! assert (d1, d0(end:-1:(end-k+1)), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A \ x; %! opts.issym = 1; opts.isreal = 1; %! d1 = eigs (fn, n, k, "sm", opts); %! assert (d1, d0(k:-1:1), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! fn = @(x) (A - 4.1 * eye (n)) \ x; %! opts.issym = 1; opts.isreal = 1; %! d1 = eigs (fn, n, k, 4.1, opts); %! assert (d1, eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! AA = speye (10); %! fn = @(x) AA * x; %! opts.issym = 1; opts.isreal = 1; %! assert (eigs (fn, 10, AA, 3, "lm", opts), [1; 1; 1], 10*eps); %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK, HAVE_UMFPACK %! [v1,d1] = eigs (A, k, "sm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "la"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "sa"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "be"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor ## Real unsymmetric tests %!shared n, k, A, d0 %! n = 20; %! k = 4; %! A = sparse ([3:n,1:n,1:(n-2)],[1:(n-2),1:n,3:n],[ones(1,n-2),1:n,-ones(1,n-2)]); %! d0 = eig (A); %! [~, idx] = sort (abs (d0)); %! d0 = d0(idx); %! rand ("state", 42); % initialize generator to make eigs behavior reproducible %!testif HAVE_ARPACK %! d1 = eigs (A, k); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k+1); %! assert (abs (d1), abs (d0(end:-1:(end-k))),1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! d1 = eigs (A, k, "sm"); %! assert (abs (d1), abs (d0(1:k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lr"); %! [~, idx] = sort (real (d0)); %! d2 = d0(idx); %! assert (real (d1), real (d2(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "sr"); %! [~, idx] = sort (real (abs (d0))); %! d2 = d0(idx); %! assert (real (d1), real (d2(1:k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "li"); %! [~, idx] = sort (imag (abs (d0))); %! d2 = d0(idx); %! assert (sort (imag (d1)), sort (imag (d2(end:-1:(end-k+1)))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "si"); %! [~, idx] = sort (imag (abs (d0))); %! d2 = d0(idx); %! assert (sort (imag (d1)), sort (imag (d2(1:k))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! d1 = eigs (A, k, 4.1); %! [~, idx0] = sort (abs (d0 - 4.1)); %! [~, idx1] = sort (abs (d1 - 4.1)); %! assert (abs (d1(idx1)), abs (d0(idx0(1:k))), 1e-11); %! assert (sort (imag (d1(idx1))), sort (imag (d0(idx0(1:k)))), 1e-11); %!testif HAVE_ARPACK, HAVE_CHOLMOD %! d1 = eigs (A, speye (n), k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! d1 = eigs (A, speye (n), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, speye (n)(q,q), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! opts.cholB = true; %! d1 = eigs (A, speye (n), k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, speye (n)(q,q), k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! assert (abs (eigs (A, k, 4.1)), abs (eigs (A, speye (n), k, 4.1)), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! assert (sort (imag (eigs (A, k, 4.1))), sort (imag (eigs (A, speye (n), k, 4.1))), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A * x; %! opts.issym = 0; opts.isreal = 1; %! d1 = eigs (fn, n, k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A \ x; %! opts.issym = 0; opts.isreal = 1; %! d1 = eigs (fn, n, k, "sm", opts); %! assert (abs (d1), d0(1:k), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! fn = @(x) (A - 4.1 * eye (n)) \ x; %! opts.issym = 0; opts.isreal = 1; %! d1 = eigs (fn, n, k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK, HAVE_UMFPACK %! [v1,d1] = eigs (A, k, "sm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lr"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "sr"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "li"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "si"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor ## Complex hermitian tests %!shared n, k, A, d0 %! n = 20; %! k = 4; %! A = sparse ([3:n,1:n,1:(n-2)],[1:(n-2),1:n,3:n],[1i*ones(1,n-2),4*ones(1,n),-1i*ones(1,n-2)]); %! d0 = eig (A); %! [~, idx] = sort (abs (d0)); %! d0 = d0(idx); %! rand ("state", 42); % initialize generator to make eigs behavior reproducible %!testif HAVE_ARPACK %! d1 = eigs (A, k); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k+1); %! assert (abs (d1), abs (d0(end:-1:(end-k))),1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! d1 = eigs (A, k, "sm"); %! assert (abs (d1), abs (d0(1:k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lr"); %! [~, idx] = sort (real (abs (d0))); %! d2 = d0(idx); %! assert (real (d1), real (d2(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "sr"); %! [~, idx] = sort (real (abs (d0))); %! d2 = d0(idx); %! assert (real (d1), real (d2(1:k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "li"); %! [~, idx] = sort (imag (abs (d0))); %! d2 = d0(idx); %! assert (sort (imag (d1)), sort (imag (d2(end:-1:(end-k+1)))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "si"); %! [~, idx] = sort (imag (abs (d0))); %! d2 = d0(idx); %! assert (sort (imag (d1)), sort (imag (d2(1:k))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! d1 = eigs (A, k, 4.1); %! [~, idx0] = sort (abs (d0 - 4.1)); %! [~, idx1] = sort (abs (d1 - 4.1)); %! assert (abs (d1(idx1)), abs (d0(idx0(1:k))), 1e-11); %! assert (sort (imag (d1(idx1))), sort (imag (d0(idx0(1:k)))), 1e-11); %!testif HAVE_ARPACK, HAVE_CHOLMOD %! d1 = eigs (A, speye (n), k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! d1 = eigs (A, speye (n), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, speye (n)(q,q), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! opts.cholB = true; %! d1 = eigs (A, speye (n), k, 4.1, opts); %! assert (abs (abs (d1)), abs (eigs (A, k, 4.1)), 1e-11); %! assert (sort (imag (abs (d1))), sort (imag (eigs (A, k, 4.1))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, speye (n)(q,q), k, 4.1, opts); %! assert (abs (abs (d1)), abs (eigs (A, k, 4.1)), 1e-11); %! assert (sort (imag (abs (d1))), sort (imag (eigs (A, k, 4.1))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! assert (abs (eigs (A, k, 4.1)), abs (eigs (A, speye (n), k, 4.1)), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! assert (sort (imag (eigs (A, k, 4.1))), sort (imag (eigs (A, speye (n), k, 4.1))), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A * x; %! opts.issym = 0; opts.isreal = 0; %! d1 = eigs (fn, n, k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A \ x; %! opts.issym = 0; opts.isreal = 0; %! d1 = eigs (fn, n, k, "sm", opts); %! assert (abs (d1), d0(1:k), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! fn = @(x) (A - 4.1 * eye (n)) \ x; %! opts.issym = 0; opts.isreal = 0; %! d1 = eigs (fn, n, k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK, HAVE_UMFPACK %! [v1,d1] = eigs (A, k, "sm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lr"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "sr"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "li"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "si"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor #### FULL MATRIX VERSIONS #### ## Real positive definite tests, n must be even %!shared n, k, A, d0, d2 %! n = 20; %! k = 4; %! A = full (sparse ([3:n,1:n,1:(n-2)],[1:(n-2),1:n,3:n],[ones(1,n-2),4*ones(1,n),ones(1,n-2)])); %! d0 = eig (A); %! d2 = sort (d0); %! [~, idx] = sort (abs (d0)); %! d0 = d0(idx); %! rand ("state", 42); % initialize generator to make eigs behavior reproducible %!testif HAVE_ARPACK %! d1 = eigs (A, k); %! assert (d1, d0(end:-1:(end-k+1)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k+1); %! assert (d1, d0(end:-1:(end-k)),1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lm"); %! assert (d1, d0(end:-1:(end-k+1)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "sm"); %! assert (d1, d0(k:-1:1), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "la"); %! assert (d1, d2(end:-1:(end-k+1)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "sa"); %! assert (d1, d2(1:k), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "be"); %! assert (d1, d2([1:floor(k/2), (end - ceil(k/2) + 1):end]), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k+1, "be"); %! assert (d1, d2([1:floor((k+1)/2), (end - ceil((k+1)/2) + 1):end]), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, 4.1); %! [~, idx0] = sort (abs (d0 - 4.1)); %! [~, idx1] = sort (abs (d1 - 4.1)); %! assert (d1(idx1), d0(idx0(1:k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, eye (n), k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! assert (eigs (A, k, 4.1), eigs (A, eye (n), k, 4.1), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! d1 = eigs (A, eye (n), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, eye (n)(q,q), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! d1 = eigs (A, eye (n), k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, eye (n)(q,q), k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! assert (eigs (A, k, 4.1), eigs (A, eye (n), k, 4.1), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A * x; %! opts.issym = 1; opts.isreal = 1; %! d1 = eigs (fn, n, k, "lm", opts); %! assert (d1, d0(end:-1:(end-k+1)), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A \ x; %! opts.issym = 1; opts.isreal = 1; %! d1 = eigs (fn, n, k, "sm", opts); %! assert (d1, d0(k:-1:1), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) (A - 4.1 * eye (n)) \ x; %! opts.issym = 1; opts.isreal = 1; %! d1 = eigs (fn, n, k, 4.1, opts); %! assert (d1, eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "sm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "la"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "sa"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "be"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor ## Real unsymmetric tests %!shared n, k, A, d0 %! n = 20; %! k = 4; %! A = full (sparse ([3:n,1:n,1:(n-2)],[1:(n-2),1:n,3:n],[ones(1,n-2),1:n,-ones(1,n-2)])); %! d0 = eig (A); %! [~, idx] = sort (abs (d0)); %! d0 = d0(idx); %! rand ("state", 42); % initialize generator to make eigs behavior reproducible %!testif HAVE_ARPACK %! d1 = eigs (A, k); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k+1); %! assert (abs (d1), abs (d0(end:-1:(end-k))),1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "sm"); %! assert (abs (d1), abs (d0(1:k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lr"); %! [~, idx] = sort (real (d0)); %! d2 = d0(idx); %! assert (real (d1), real (d2(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "sr"); %! [~, idx] = sort (real (abs (d0))); %! d2 = d0(idx); %! assert (real (d1), real (d2(1:k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "li"); %! [~, idx] = sort (imag (abs (d0))); %! d2 = d0(idx); %! assert (sort (imag (d1)), sort (imag (d2(end:-1:(end-k+1)))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "si"); %! [~, idx] = sort (imag (abs (d0))); %! d2 = d0(idx); %! assert (sort (imag (d1)), sort (imag (d2(1:k))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, 4.1); %! [~, idx0] = sort (abs (d0 - 4.1)); %! [~, idx1] = sort (abs (d1 - 4.1)); %! assert (abs (d1(idx1)), abs (d0(idx0(1:k))), 1e-11); %! assert (sort (imag (d1(idx1))), sort (imag (d0(idx0(1:k)))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, eye (n), k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! d1 = eigs (A, eye (n), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, eye (n)(q,q), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! d1 = eigs (A, eye (n), k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, eye (n)(q,q), k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! assert (abs (eigs (A, k, 4.1)), abs (eigs (A, eye (n), k, 4.1)), 1e-11); %!testif HAVE_ARPACK %! assert (sort (imag (eigs (A, k, 4.1))), sort (imag (eigs (A, eye (n), k, 4.1))), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A * x; %! opts.issym = 0; opts.isreal = 1; %! d1 = eigs (fn, n, k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A \ x; %! opts.issym = 0; opts.isreal = 1; %! d1 = eigs (fn, n, k, "sm", opts); %! assert (abs (d1), d0(1:k), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) (A - 4.1 * eye (n)) \ x; %! opts.issym = 0; opts.isreal = 1; %! d1 = eigs (fn, n, k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "sm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lr"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "sr"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "li"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "si"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor ## Complex hermitian tests %!shared n, k, A, d0 %! n = 20; %! k = 4; %! A = full (sparse ([3:n,1:n,1:(n-2)],[1:(n-2),1:n,3:n],[1i*ones(1,n-2),4*ones(1,n),-1i*ones(1,n-2)])); %! d0 = eig (A); %! [~, idx] = sort (abs (d0)); %! d0 = d0(idx); %! rand ("state", 42); % initialize generator to make eigs behavior reproducible %!testif HAVE_ARPACK %! d1 = eigs (A, k); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k+1); %! assert (abs (d1), abs (d0(end:-1:(end-k))),1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "sm"); %! assert (abs (d1), abs (d0(1:k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lr"); %! [~, idx] = sort (real (abs (d0))); %! d2 = d0(idx); %! assert (real (d1), real (d2(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "sr"); %! [~, idx] = sort (real (abs (d0))); %! d2 = d0(idx); %! assert (real (d1), real (d2(1:k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "li"); %! [~, idx] = sort (imag (abs (d0))); %! d2 = d0(idx); %! assert (sort (imag (d1)), sort (imag (d2(end:-1:(end-k+1)))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "si"); %! [~, idx] = sort (imag (abs (d0))); %! d2 = d0(idx); %! assert (sort (imag (d1)), sort (imag (d2(1:k))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, 4.1); %! [~, idx0] = sort (abs (d0 - 4.1)); %! [~, idx1] = sort (abs (d1 - 4.1)); %! assert (abs (d1(idx1)), abs (d0(idx0(1:k))), 1e-11); %! assert (sort (imag (d1(idx1))), sort (imag (d0(idx0(1:k)))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, eye (n), k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! d1 = eigs (A, eye (n), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, eye (n)(q,q), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! d1 = eigs (A, eye (n), k, 4.1, opts); %! assert (abs (abs (d1)), abs (eigs (A, k, 4.1)), 1e-11); %! assert (sort (imag (abs (d1))), sort (imag (eigs (A, k, 4.1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, eye (n)(q,q), k, 4.1, opts); %! assert (abs (abs (d1)), abs (eigs (A, k, 4.1)), 1e-11); %! assert (sort (imag (abs (d1))), sort (imag (eigs (A, k, 4.1))), 1e-11); %!testif HAVE_ARPACK %! assert (abs (eigs (A, k, 4.1)), abs (eigs (A, eye (n), k, 4.1)), 1e-11); %!testif HAVE_ARPACK %! assert (sort (imag (eigs (A, k, 4.1))), sort (imag (eigs (A, eye (n), k, 4.1))), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A * x; %! opts.issym = 0; opts.isreal = 0; %! d1 = eigs (fn, n, k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A \ x; %! opts.issym = 0; opts.isreal = 0; %! d1 = eigs (fn, n, k, "sm", opts); %! assert (abs (d1), d0(1:k), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) (A - 4.1 * eye (n)) \ x; %! opts.issym = 0; opts.isreal = 0; %! d1 = eigs (fn, n, k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "sm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lr"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "sr"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "li"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "si"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!test %! A = 2 * diag (ones (10, 1)) - diag (ones (9, 1), 1) - diag (ones (9, 1), -1); %! B = diag (ones (10, 1)); %! reseig = eig (A, B); %! [~, idx] = sort (abs (reseig), "ascend"); %! assert (eigs (A, B, 10, 0), reseig (idx)) %!test %! X = [70 47 42 39 50 73 79 23; %! 19 52 61 80 36 76 63 68; %! 14 34 66 65 29 4 72 9; %! 24 8 78 49 58 54 43 33; %! 62 69 32 31 40 46 22 28; %! 48 12 45 59 10 17 15 25; %! 64 67 77 56 13 55 41 74; %! 37 38 18 21 11 3 71 7; %! 5 35 16 1 51 27 26 44; %! 30 57 60 75 2 53 20 6]; %! Z = X * X'; %! r = rank (Z); %! assert (r, 8); %! [V, D] = eigs (Z, r, "lm"); %! ZZ = V * D * V'; %! tmp = abs (Z - ZZ); %! assert (max (tmp(:)) < 5e-11); %!assert (eigs (diag (1:5), 5, "sa"), [1;2;3;4;5]); %!assert (eigs (diag (1:5), 5, "la"), [5;4;3;2;1]); %!assert (eigs (diag (1:5), 3, "be"), [1;4;5]);