Mercurial > octave-nkf
annotate doc/interpreter/quad.txi @ 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> |
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date | Tue, 20 Jan 2015 08:26:57 -0500 |
parents | d63878346099 |
children | 0e1f5a750d00 |
rev | line source |
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1 @c Copyright (C) 1996-2013 John W. Eaton |
7018 | 2 @c |
3 @c This file is part of Octave. | |
4 @c | |
5 @c Octave is free software; you can redistribute it and/or modify it | |
6 @c under the terms of the GNU General Public License as published by the | |
7 @c Free Software Foundation; either version 3 of the License, or (at | |
8 @c your option) any later version. | |
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9 @c |
7018 | 10 @c Octave is distributed in the hope that it will be useful, but WITHOUT |
11 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License | |
13 @c for more details. | |
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14 @c |
7018 | 15 @c You should have received a copy of the GNU General Public License |
16 @c along with Octave; see the file COPYING. If not, see | |
17 @c <http://www.gnu.org/licenses/>. | |
3294 | 18 |
6741 | 19 @node Numerical Integration |
20 @chapter Numerical Integration | |
21 | |
22 Octave comes with several built-in functions for computing the integral | |
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23 of a function numerically (termed quadrature). These functions all solve |
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24 1-dimensional integration problems. |
3294 | 25 |
26 @menu | |
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27 * Functions of One Variable:: |
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28 * Orthogonal Collocation:: |
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29 * Functions of Multiple Variables:: |
3294 | 30 @end menu |
31 | |
4167 | 32 @node Functions of One Variable |
3294 | 33 @section Functions of One Variable |
34 | |
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35 Octave supports five different algorithms for computing the integral |
6741 | 36 @tex |
37 $$ | |
38 \int_a^b f(x) d x | |
39 $$ | |
40 @end tex | |
41 of a function @math{f} over the interval from @math{a} to @math{b}. | |
42 These are | |
43 | |
44 @table @code | |
45 @item quad | |
46 Numerical integration based on Gaussian quadrature. | |
47 | |
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48 @item quadv |
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49 Numerical integration using an adaptive vectorized Simpson's rule. |
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50 |
6741 | 51 @item quadl |
52 Numerical integration using an adaptive Lobatto rule. | |
53 | |
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54 @item quadgk |
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55 Numerical integration using an adaptive Gauss-Konrod rule. |
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56 |
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57 @item quadcc |
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58 Numerical integration using adaptive Clenshaw-Curtis rules. |
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59 |
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60 @item trapz, cumtrapz |
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61 Numerical integration of data using the trapezoidal method. |
6741 | 62 @end table |
63 | |
64 @noindent | |
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65 The best quadrature algorithm to use depends on the integrand. If you have |
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66 empirical data, rather than a function, the choice is @code{trapz} or |
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67 @code{cumtrapz}. If you are uncertain about the characteristics of the |
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68 integrand, @code{quadcc} will be the most robust as it can handle |
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69 discontinuities, singularities, oscillatory functions, and infinite intervals. |
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70 When the integrand is smooth @code{quadgk} may be the fastest of the |
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71 algorithms. |
6741 | 72 |
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73 @multitable @columnfractions 0.05 0.15 .80 |
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74 @headitem @tab Function @tab Characteristics |
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75 @item @tab quad @tab Low accuracy with nonsmooth integrands |
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76 @item @tab quadv @tab Medium accuracy with smooth integrands |
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77 @item @tab quadl @tab Medium accuracy with smooth integrands. Slower than quadgk. |
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78 @item @tab quadgk @tab Medium accuracy (@math{1e^{-6}}--@math{1e^{-9}}) with smooth integrands. |
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79 @item @tab @tab Handles oscillatory functions and infinite bounds |
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80 @item @tab quadcc @tab Low to High accuracy with nonsmooth/smooth integrands |
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81 @item @tab @tab Handles oscillatory functions, singularities, and infinite bounds |
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82 @end multitable |
3294 | 83 |
84 | |
85 Here is an example of using @code{quad} to integrate the function | |
86 @tex | |
87 $$ | |
88 f(x) = x \sin (1/x) \sqrt {|1 - x|} | |
89 $$ | |
90 from $x = 0$ to $x = 3$. | |
91 @end tex | |
6741 | 92 @ifnottex |
3294 | 93 |
94 @example | |
95 @var{f}(@var{x}) = @var{x} * sin (1/@var{x}) * sqrt (abs (1 - @var{x})) | |
96 @end example | |
97 | |
98 @noindent | |
99 from @var{x} = 0 to @var{x} = 3. | |
6741 | 100 @end ifnottex |
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102 This is a fairly difficult integration (plot the function over the range | |
103 of integration to see why). | |
104 | |
105 The first step is to define the function: | |
106 | |
107 @example | |
108 @group | |
109 function y = f (x) | |
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110 y = x .* sin (1./x) .* sqrt (abs (1 - x)); |
3294 | 111 endfunction |
112 @end group | |
113 @end example | |
114 | |
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115 Note the use of the `dot' forms of the operators. This is not necessary for |
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116 the @code{quad} integrator, but is required by the other integrators. In any |
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117 case, it makes it much easier to generate a set of points for plotting because |
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118 it is possible to call the function with a vector argument to produce a vector |
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119 result. |
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121 The second step is to call quad with the limits of integration: |
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123 @example | |
124 @group | |
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125 [q, ier, nfun, err] = quad ("f", 0, 3) |
3294 | 126 @result{} 1.9819 |
127 @result{} 1 | |
128 @result{} 5061 | |
129 @result{} 1.1522e-07 | |
130 @end group | |
131 @end example | |
132 | |
133 Although @code{quad} returns a nonzero value for @var{ier}, the result | |
134 is reasonably accurate (to see why, examine what happens to the result | |
135 if you move the lower bound to 0.1, then 0.01, then 0.001, etc.). | |
136 | |
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137 The function @qcode{"f"} can be the string name of a function, a function |
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138 handle, or an inline function. These options make it quite easy to do |
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139 integration without having to fully define a function in an m-file. For |
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140 example: |
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141 |
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142 @example |
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143 @group |
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144 # Verify integral (x^3) = x^4/4 |
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145 f = inline ("x.^3"); |
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146 quadgk (f, 0, 1) |
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147 @result{} 0.25000 |
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148 |
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149 # Verify gamma function = (n-1)! for n = 4 |
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150 f = @@(x) x.^3 .* exp (-x); |
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151 quadcc (f, 0, Inf) |
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152 @result{} 6.0000 |
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153 @end group |
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154 @end example |
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155 |
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156 @DOCSTRING(quad) |
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157 |
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158 @DOCSTRING(quad_options) |
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159 |
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160 @DOCSTRING(quadv) |
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161 |
6502 | 162 @DOCSTRING(quadl) |
163 | |
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164 @DOCSTRING(quadgk) |
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165 |
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166 @DOCSTRING(quadcc) |
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167 |
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168 Sometimes one does not have the function, but only the raw (x, y) points from |
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169 which to perform an integration. This can occur when collecting data in an |
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170 experiment. The @code{trapz} function can integrate these values as shown in |
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171 the following example where "data" has been collected on the cosine function |
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172 over the range [0, pi/2). |
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173 |
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174 @example |
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175 @group |
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176 x = 0:0.1:pi/2; # Uniformly spaced points |
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177 y = cos (x); |
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178 trapz (x, y) |
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179 @result{} 0.99666 |
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180 @end group |
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181 @end example |
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182 |
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183 The answer is reasonably close to the exact value of 1. Ordinary quadrature |
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184 is sensitive to the characteristics of the integrand. Empirical integration |
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185 depends not just on the integrand, but also on the particular points chosen to |
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186 represent the function. Repeating the example above with the sine function |
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187 over the range [0, pi/2) produces far inferior results. |
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188 |
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189 @example |
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190 @group |
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191 x = 0:0.1:pi/2; # Uniformly spaced points |
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192 y = sin (x); |
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193 trapz (x, y) |
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194 @result{} 0.92849 |
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195 @end group |
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196 @end example |
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197 |
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198 However, a slightly different choice of data points can change the result |
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199 significantly. The same integration, with the same number of points, but |
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200 spaced differently produces a more accurate answer. |
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201 |
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202 @example |
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203 @group |
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204 x = linspace (0, pi/2, 16); # Uniformly spaced, but including endpoint |
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205 y = sin (x); |
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206 trapz (x, y) |
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207 @result{} 0.99909 |
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208 @end group |
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209 @end example |
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210 |
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211 In general there may be no way of knowing the best distribution of points ahead |
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212 of time. Or the points may come from an experiment where there is no freedom to |
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213 select the best distribution. In any case, one must remain aware of this |
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214 issue when using @code{trapz}. |
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215 |
6502 | 216 @DOCSTRING(trapz) |
217 | |
218 @DOCSTRING(cumtrapz) | |
219 | |
4167 | 220 @node Orthogonal Collocation |
3294 | 221 @section Orthogonal Collocation |
222 | |
3368 | 223 @DOCSTRING(colloc) |
3294 | 224 |
225 Here is an example of using @code{colloc} to generate weight matrices | |
226 for solving the second order differential equation | |
227 @tex | |
228 $u^\prime - \alpha u^{\prime\prime} = 0$ with the boundary conditions | |
229 $u(0) = 0$ and $u(1) = 1$. | |
230 @end tex | |
6741 | 231 @ifnottex |
3294 | 232 @var{u}' - @var{alpha} * @var{u}'' = 0 with the boundary conditions |
233 @var{u}(0) = 0 and @var{u}(1) = 1. | |
6741 | 234 @end ifnottex |
3294 | 235 |
236 First, we can generate the weight matrices for @var{n} points (including | |
237 the endpoints of the interval), and incorporate the boundary conditions | |
238 in the right hand side (for a specific value of | |
239 @tex | |
240 $\alpha$). | |
241 @end tex | |
6741 | 242 @ifnottex |
3294 | 243 @var{alpha}). |
6741 | 244 @end ifnottex |
3294 | 245 |
246 @example | |
247 @group | |
248 n = 7; | |
249 alpha = 0.1; | |
250 [r, a, b] = colloc (n-2, "left", "right"); | |
251 at = a(2:n-1,2:n-1); | |
252 bt = b(2:n-1,2:n-1); | |
253 rhs = alpha * b(2:n-1,n) - a(2:n-1,n); | |
254 @end group | |
255 @end example | |
256 | |
257 Then the solution at the roots @var{r} is | |
258 | |
259 @example | |
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260 @group |
3294 | 261 u = [ 0; (at - alpha * bt) \ rhs; 1] |
262 @result{} [ 0.00; 0.004; 0.01 0.00; 0.12; 0.62; 1.00 ] | |
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263 @end group |
3294 | 264 @end example |
6741 | 265 |
266 @node Functions of Multiple Variables | |
267 @section Functions of Multiple Variables | |
268 | |
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269 Octave does not have built-in functions for computing the integral of |
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270 functions of multiple variables directly. It is possible, however, to |
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271 compute the integral of a function of multiple variables using the |
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272 existing functions for one-dimensional integrals. |
6741 | 273 |
274 To illustrate how the integration can be performed, we will integrate | |
275 the function | |
276 @tex | |
277 $$ | |
278 f(x, y) = \sin(\pi x y)\sqrt{x y} | |
279 $$ | |
280 @end tex | |
281 @ifnottex | |
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282 |
6741 | 283 @example |
284 f(x, y) = sin(pi*x*y)*sqrt(x*y) | |
285 @end example | |
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286 |
6741 | 287 @end ifnottex |
288 for @math{x} and @math{y} between 0 and 1. | |
289 | |
290 The first approach creates a function that integrates @math{f} with | |
291 respect to @math{x}, and then integrates that function with respect to | |
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292 @math{y}. Because @code{quad} is written in Fortran it cannot be called |
6741 | 293 recursively. This means that @code{quad} cannot integrate a function |
294 that calls @code{quad}, and hence cannot be used to perform the double | |
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295 integration. Any of the other integrators, however, can be used which is |
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296 what the following code demonstrates. |
6741 | 297 |
298 @example | |
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299 @group |
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300 function q = g(y) |
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301 q = ones (size (y)); |
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302 for i = 1:length (y) |
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303 f = @@(x) sin (pi*x.*y(i)) .* sqrt (x.*y(i)); |
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304 q(i) = quadgk (f, 0, 1); |
6741 | 305 endfor |
306 endfunction | |
307 | |
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308 I = quadgk ("g", 0, 1) |
6741 | 309 @result{} 0.30022 |
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310 @end group |
6741 | 311 @end example |
312 | |
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313 The above process can be simplified with the @code{dblquad} and |
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314 @code{triplequad} functions for integrals over two and three |
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315 variables. For example: |
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316 |
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317 @example |
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318 @group |
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319 I = dblquad (@@(x, y) sin (pi*x.*y) .* sqrt (x.*y), 0, 1, 0, 1) |
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320 @result{} 0.30022 |
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321 @end group |
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322 @end example |
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323 |
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324 @DOCSTRING(dblquad) |
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325 |
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326 @DOCSTRING(triplequad) |
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327 |
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328 The above mentioned approach works, but is fairly slow, and that problem |
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329 increases exponentially with the dimensionality of the integral. Another |
6741 | 330 possible solution is to use Orthogonal Collocation as described in the |
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331 previous section (@pxref{Orthogonal Collocation}). The integral of a function |
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332 @math{f(x,y)} for @math{x} and @math{y} between 0 and 1 can be approximated |
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333 using @math{n} points by |
6741 | 334 @tex |
335 $$ | |
336 \int_0^1 \int_0^1 f(x,y) d x d y \approx \sum_{i=1}^n \sum_{j=1}^n q_i q_j f(r_i, r_j), | |
337 $$ | |
338 @end tex | |
339 @ifnottex | |
340 the sum over @code{i=1:n} and @code{j=1:n} of @code{q(i)*q(j)*f(r(i),r(j))}, | |
341 @end ifnottex | |
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342 where @math{q} and @math{r} is as returned by @code{colloc (n)}. The |
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343 generalization to more than two variables is straight forward. The |
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344 following code computes the studied integral using @math{n=8} points. |
6741 | 345 |
346 @example | |
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347 @group |
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348 f = @@(x,y) sin (pi*x*y') .* sqrt (x*y'); |
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349 n = 8; |
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350 [t, ~, ~, q] = colloc (n); |
6741 | 351 I = q'*f(t,t)*q; |
352 @result{} 0.30022 | |
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353 @end group |
6741 | 354 @end example |
355 | |
356 @noindent | |
357 It should be noted that the number of points determines the quality | |
358 of the approximation. If the integration needs to be performed between | |
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359 @math{a} and @math{b}, instead of 0 and 1, then a change of variables is needed. |
6741 | 360 |