Mercurial > octave-nkf
view scripts/sparse/pcg.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 | d63878346099 |
children | 0e1f5a750d00 |
line wrap: on
line source
## Copyright (C) 2004-2013 Piotr Krzyzanowski ## ## 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{x} =} pcg (@var{A}, @var{b}, @var{tol}, @var{maxit}, @var{m1}, @var{m2}, @var{x0}, @dots{}) ## @deftypefnx {Function File} {[@var{x}, @var{flag}, @var{relres}, @var{iter}, @var{resvec}, @var{eigest}] =} pcg (@dots{}) ## ## Solve the linear system of equations @w{@code{@var{A} * @var{x} = @var{b}}} ## by means of the Preconditioned Conjugate Gradient iterative method. The ## input arguments are ## ## @itemize ## @item ## @var{A} can be either a square (preferably sparse) matrix or a function ## handle, inline function or string containing the name of a function which ## computes @w{@code{@var{A} * @var{x}}}. In principle, @var{A} should be ## symmetric and positive definite; if @code{pcg} finds @var{A} not to be ## positive definite, a warning is printed and the @var{flag} output will be ## set. ## ## @item ## @var{b} is the right-hand side vector. ## ## @item ## @var{tol} is the required relative tolerance for the residual error, ## @w{@code{@var{b} - @var{A} * @var{x}}}. The iteration stops if ## @w{@code{norm (@var{b} - @var{A} * @var{x})} @leq{} ## @w{@var{tol} * norm (@var{b})}}. ## If @var{tol} is omitted or empty then a tolerance of 1e-6 is used. ## ## @item ## @var{maxit} is the maximum allowable number of iterations; if @var{maxit} ## is omitted or empty then a value of 20 is used. ## ## @item ## @var{m} = @var{m1} * @var{m2} is the (left) preconditioning matrix, so that ## the iteration is (theoretically) equivalent to solving by @code{pcg} ## @w{@code{@var{P} * @var{x} = @var{m} \ @var{b}}}, with ## @w{@code{@var{P} = @var{m} \ @var{A}}}. ## Note that a proper choice of the preconditioner may dramatically ## improve the overall performance of the method. Instead of matrices ## @var{m1} and @var{m2}, the user may pass two functions which return ## the results of applying the inverse of @var{m1} and @var{m2} to ## a vector (usually this is the preferred way of using the preconditioner). ## If @var{m1} is omitted or empty @code{[]} then no preconditioning is ## applied. If @var{m2} is omitted, @var{m} = @var{m1} will be used as ## a preconditioner. ## ## @item ## @var{x0} is the initial guess. If @var{x0} is omitted or empty then the ## function sets @var{x0} to a zero vector by default. ## @end itemize ## ## The arguments which follow @var{x0} are treated as parameters, and passed in ## a proper way to any of the functions (@var{A} or @var{m}) which are passed ## to @code{pcg}. See the examples below for further details. The output ## arguments are ## ## @itemize ## @item ## @var{x} is the computed approximation to the solution of ## @w{@code{@var{A} * @var{x} = @var{b}}}. ## ## @item ## @var{flag} reports on the convergence. A value of 0 means the solution ## converged and the tolerance criterion given by @var{tol} is satisfied. ## A value of 1 means that the @var{maxit} limit for the iteration count was ## reached. A value of 3 indicates that the (preconditioned) matrix was found ## not to be positive definite. ## ## @item ## @var{relres} is the ratio of the final residual to its initial value, ## measured in the Euclidean norm. ## ## @item ## @var{iter} is the actual number of iterations performed. ## ## @item ## @var{resvec} describes the convergence history of the method. ## @code{@var{resvec}(i,1)} is the Euclidean norm of the residual, and ## @code{@var{resvec}(i,2)} is the preconditioned residual norm, after the ## (@var{i}-1)-th iteration, @code{@var{i} = 1, 2, @dots{}, @var{iter}+1}. ## The preconditioned residual norm is defined as ## @code{norm (@var{r}) ^ 2 = @var{r}' * (@var{m} \ @var{r})} where ## @code{@var{r} = @var{b} - @var{A} * @var{x}}, see also the ## description of @var{m}. If @var{eigest} is not required, only ## @code{@var{resvec}(:,1)} is returned. ## ## @item ## @var{eigest} returns the estimate for the smallest @code{@var{eigest}(1)} ## and largest @code{@var{eigest}(2)} eigenvalues of the preconditioned matrix ## @w{@code{@var{P} = @var{m} \ @var{A}}}. In particular, if no ## preconditioning is used, the estimates for the extreme eigenvalues of ## @var{A} are returned. @code{@var{eigest}(1)} is an overestimate and ## @code{@var{eigest}(2)} is an underestimate, so that ## @code{@var{eigest}(2) / @var{eigest}(1)} is a lower bound for ## @code{cond (@var{P}, 2)}, which nevertheless in the limit should ## theoretically be equal to the actual value of the condition number. ## The method which computes @var{eigest} works only for symmetric positive ## definite @var{A} and @var{m}, and the user is responsible for verifying this ## assumption. ## @end itemize ## ## Let us consider a trivial problem with a diagonal matrix (we exploit the ## sparsity of A) ## ## @example ## @group ## n = 10; ## A = diag (sparse (1:n)); ## b = rand (n, 1); ## [l, u, p, q] = luinc (A, 1.e-3); ## @end group ## @end example ## ## @sc{Example 1:} Simplest use of @code{pcg} ## ## @example ## x = pcg (A, b) ## @end example ## ## @sc{Example 2:} @code{pcg} with a function which computes ## @code{@var{A} * @var{x}} ## ## @example ## @group ## function y = apply_a (x) ## y = [1:N]' .* x; ## endfunction ## ## x = pcg ("apply_a", b) ## @end group ## @end example ## ## @sc{Example 3:} @code{pcg} with a preconditioner: @var{l} * @var{u} ## ## @example ## x = pcg (A, b, 1.e-6, 500, l*u) ## @end example ## ## @sc{Example 4:} @code{pcg} with a preconditioner: @var{l} * @var{u}. ## Faster than @sc{Example 3} since lower and upper triangular matrices ## are easier to invert ## ## @example ## x = pcg (A, b, 1.e-6, 500, l, u) ## @end example ## ## @sc{Example 5:} Preconditioned iteration, with full diagnostics. The ## preconditioner (quite strange, because even the original matrix ## @var{A} is trivial) is defined as a function ## ## @example ## @group ## function y = apply_m (x) ## k = floor (length (x) - 2); ## y = x; ## y(1:k) = x(1:k) ./ [1:k]'; ## endfunction ## ## [x, flag, relres, iter, resvec, eigest] = ... ## pcg (A, b, [], [], "apply_m"); ## semilogy (1:iter+1, resvec); ## @end group ## @end example ## ## @sc{Example 6:} Finally, a preconditioner which depends on a ## parameter @var{k}. ## ## @example ## @group ## function y = apply_M (x, varargin) ## K = varargin@{1@}; ## y = x; ## y(1:K) = x(1:K) ./ [1:K]'; ## endfunction ## ## [x, flag, relres, iter, resvec, eigest] = ... ## pcg (A, b, [], [], "apply_m", [], [], 3) ## @end group ## @end example ## ## References: ## ## @enumerate ## @item ## C.T. Kelley, @cite{Iterative Methods for Linear and Nonlinear Equations}, ## SIAM, 1995. (the base PCG algorithm) ## ## @item ## Y. Saad, @cite{Iterative Methods for Sparse Linear Systems}, ## @nospell{PWS} 1996. (condition number estimate from PCG) ## Revised version of this book is available online at ## @url{http://www-users.cs.umn.edu/~saad/books.html} ## @end enumerate ## ## @seealso{sparse, pcr} ## @end deftypefn ## Author: Piotr Krzyzanowski <piotr.krzyzanowski@mimuw.edu.pl> ## Modified by: Vittoria Rezzonico <vittoria.rezzonico@epfl.ch> ## - Add the ability to provide the pre-conditioner as two separate matrices function [x, flag, relres, iter, resvec, eigest] = pcg (A, b, tol, maxit, m1, m2, x0, varargin) ## M = M1*M2 if (nargin < 7 || isempty (x0)) x = zeros (size (b)); else x = x0; endif if (nargin < 5 || isempty (m1)) exist_m1 = 0; else exist_m1 = 1; endif if (nargin < 6 || isempty (m2)) exist_m2 = 0; else exist_m2 = 1; endif if (nargin < 4 || isempty (maxit)) maxit = min (rows (b), 20); endif maxit += 2; if (nargin < 3 || isempty (tol)) tol = 1e-6; endif preconditioned_residual_out = false; if (nargout > 5) T = zeros (maxit, maxit); preconditioned_residual_out = true; endif ## Assume A is positive definite. matrix_positive_definite = true; p = zeros (size (b)); oldtau = 1; if (isnumeric (A)) ## A is a matrix. r = b - A*x; else ## A should be a function. r = b - feval (A, x, varargin{:}); endif b_norm = norm (b); resvec(1,1) = norm (r); alpha = 1; iter = 2; while (resvec(iter-1,1) > tol * b_norm && iter < maxit) if (exist_m1) if (isnumeric (m1)) y = m1 \ r; else y = feval (m1, r, varargin{:}); endif else y = r; endif if (exist_m2) if (isnumeric (m2)) z = m2 \ y; else z = feval (m2, y, varargin{:}); endif else z = y; endif tau = z' * r; resvec(iter-1,2) = sqrt (tau); beta = tau / oldtau; oldtau = tau; p = z + beta * p; if (isnumeric (A)) ## A is a matrix. w = A * p; else ## A should be a function. w = feval (A, p, varargin{:}); endif ## Needed only for eigest. oldalpha = alpha; alpha = tau / (p'*w); if (alpha <= 0.0) ## Negative matrix. matrix_positive_definite = false; endif x += alpha * p; r -= alpha * w; if (nargout > 5 && iter > 2) T(iter-1:iter, iter-1:iter) = T(iter-1:iter, iter-1:iter) + ... [1 sqrt(beta); sqrt(beta) beta]./oldalpha; ## EVS = eig (T(2:iter-1,2:iter-1)); ## fprintf (stderr,"PCG condest: %g (iteration: %d)\n", max (EVS)/min (EVS),iter); endif resvec(iter,1) = norm (r); iter++; endwhile if (nargout > 5) if (matrix_positive_definite) if (iter > 3) T = T(2:iter-2,2:iter-2); l = eig (T); eigest = [min(l), max(l)]; ## fprintf (stderr, "pcg condest: %g\n", eigest(2)/eigest(1)); else eigest = [NaN, NaN]; warning ("pcg: eigenvalue estimate failed: iteration converged too fast"); endif else eigest = [NaN, NaN]; endif ## Apply the preconditioner once more and finish with the precond ## residual. if (exist_m1) if (isnumeric (m1)) y = m1 \ r; else y = feval (m1, r, varargin{:}); endif else y = r; endif if (exist_m2) if (isnumeric (m2)) z = m2 \ y; else z = feval (m2, y, varargin{:}); endif else z = y; endif resvec(iter-1,2) = sqrt (r' * z); else resvec = resvec(:,1); endif flag = 0; relres = resvec(iter-1,1) ./ resvec(1,1); iter -= 2; if (iter >= maxit - 2) flag = 1; if (nargout < 2) warning ("pcg: maximum number of iterations (%d) reached\n", iter); warning ("the initial residual norm was reduced %g times.\n", ... 1.0 / relres); endif elseif (nargout < 2) fprintf (stderr, "pcg: converged in %d iterations. ", iter); fprintf (stderr, "the initial residual norm was reduced %g times.\n",... 1.0/relres); endif if (! matrix_positive_definite) flag = 3; if (nargout < 2) warning ("pcg: matrix not positive definite?\n"); endif endif endfunction %!demo %! ## Simplest usage of pcg (see also 'help pcg') %! %! N = 10; %! A = diag ([1:N]); b = rand (N, 1); %! y = A \ b; # y is the true solution %! x = pcg (A, b); %! printf ("The solution relative error is %g\n", norm (x - y) / norm (y)); %! %! ## You shouldn't be afraid if pcg issues some warning messages in this %! ## example: watch out in the second example, why it takes N iterations %! ## of pcg to converge to (a very accurate, by the way) solution %!demo %! ## Full output from pcg, except for the eigenvalue estimates %! ## We use this output to plot the convergence history %! %! N = 10; %! A = diag ([1:N]); b = rand (N, 1); %! X = A \ b; # X is the true solution %! [x, flag, relres, iter, resvec] = pcg (A, b); %! printf ("The solution relative error is %g\n", norm (x - X) / norm (X)); %! title ("Convergence history"); %! semilogy ([0:iter], resvec / resvec(1), "o-g"); %! xlabel ("Iteration"); ylabel ("log(||b-Ax||/||b||)"); %! legend ("relative residual"); %!demo %! ## Full output from pcg, including the eigenvalue estimates %! ## Hilbert matrix is extremely ill-conditioned, so pcg WILL have problems %! %! N = 10; %! A = hilb (N); b = rand (N, 1); %! X = A \ b; # X is the true solution %! [x, flag, relres, iter, resvec, eigest] = pcg (A, b, [], 200); %! printf ("The solution relative error is %g\n", norm (x - X) / norm (X)); %! printf ("Condition number estimate is %g\n", eigest(2) / eigest(1)); %! printf ("Actual condition number is %g\n", cond (A)); %! title ("Convergence history"); %! semilogy ([0:iter], resvec, ["o-g";"+-r"]); %! xlabel ("Iteration"); ylabel ("log(||b-Ax||)"); %! legend ("absolute residual", "absolute preconditioned residual"); %!demo %! ## Full output from pcg, including the eigenvalue estimates %! ## We use the 1-D Laplacian matrix for A, and cond(A) = O(N^2) %! ## and that's the reason we need some preconditioner; here we take %! ## a very simple and not powerful Jacobi preconditioner, %! ## which is the diagonal of A. %! %! N = 100; %! A = zeros (N, N); %! for i = 1 : N - 1 # form 1-D Laplacian matrix %! A(i:i+1, i:i+1) = [2 -1; -1 2]; %! endfor %! b = rand (N, 1); %! X = A \ b; # X is the true solution %! maxit = 80; %! printf ("System condition number is %g\n", cond (A)); %! ## No preconditioner: the convergence is very slow! %! %! [x, flag, relres, iter, resvec, eigest] = pcg (A, b, [], maxit); %! printf ("System condition number estimate is %g\n", eigest(2) / eigest(1)); %! title ("Convergence history"); %! semilogy ([0:iter], resvec(:,1), "o-g"); %! xlabel ("Iteration"); ylabel ("log(||b-Ax||)"); %! legend ("NO preconditioning: absolute residual"); %! %! pause (1); %! ## Test Jacobi preconditioner: it will not help much!!! %! %! M = diag (diag (A)); # Jacobi preconditioner %! [x, flag, relres, iter, resvec, eigest] = pcg (A, b, [], maxit, M); %! printf ("JACOBI preconditioned system condition number estimate is %g\n", eigest(2) / eigest(1)); %! hold on; %! semilogy ([0:iter], resvec(:,1), "o-r"); %! legend ("NO preconditioning: absolute residual", ... %! "JACOBI preconditioner: absolute residual"); %! %! pause (1); %! ## Test nonoverlapping block Jacobi preconditioner: it will help much! %! %! M = zeros (N, N); k = 4; %! for i = 1 : k : N # form 1-D Laplacian matrix %! M(i:i+k-1, i:i+k-1) = A(i:i+k-1, i:i+k-1); %! endfor %! [x, flag, relres, iter, resvec, eigest] = pcg (A, b, [], maxit, M); %! printf ("BLOCK JACOBI preconditioned system condition number estimate is %g\n", eigest(2) / eigest(1)); %! semilogy ([0:iter], resvec(:,1), "o-b"); %! legend ("NO preconditioning: absolute residual", ... %! "JACOBI preconditioner: absolute residual", ... %! "BLOCK JACOBI preconditioner: absolute residual"); %! hold off; %!test %! ## solve small diagonal system %! %! N = 10; %! A = diag ([1:N]); b = rand (N, 1); %! X = A \ b; # X is the true solution %! [x, flag] = pcg (A, b, [], N+1); %! assert (norm (x - X) / norm (X), 0, 1e-10); %! assert (flag, 0); %!test %! ## solve small indefinite diagonal system %! ## despite A is indefinite, the iteration continues and converges %! ## indefiniteness of A is detected %! %! N = 10; %! A = diag([1:N] .* (-ones(1, N) .^ 2)); b = rand (N, 1); %! X = A \ b; # X is the true solution %! [x, flag] = pcg (A, b, [], N+1); %! assert (norm (x - X) / norm (X), 0, 1e-10); %! assert (flag, 3); %!test %! ## solve tridiagonal system, do not converge in default 20 iterations %! %! N = 100; %! A = zeros (N, N); %! for i = 1 : N - 1 # form 1-D Laplacian matrix %! A(i:i+1, i:i+1) = [2 -1; -1 2]; %! endfor %! b = ones (N, 1); %! X = A \ b; # X is the true solution %! [x, flag, relres, iter, resvec, eigest] = pcg (A, b, 1e-12); %! assert (flag); %! assert (relres > 1.0); %! assert (iter, 20); # should perform max allowable default number of iterations %!test %! ## solve tridiagonal system with 'perfect' preconditioner %! ## which converges in one iteration, so the eigest does not %! ## work and issues a warning %! %! N = 100; %! A = zeros (N, N); %! for i = 1 : N - 1 # form 1-D Laplacian matrix %! A (i:i+1, i:i+1) = [2 -1; -1 2]; %! endfor %! b = ones (N, 1); %! X = A \ b; # X is the true solution %! [x, flag, relres, iter, resvec, eigest] = pcg (A, b, [], [], A, [], b); %! assert (norm (x - X) / norm (X), 0, 1e-6); %! assert (flag, 0); %! assert (iter, 1); # should converge in one iteration %! assert (isnan (eigest), isnan ([NaN, NaN]));