Mercurial > hg > octave-nkf
annotate scripts/specfun/expint.m @ 19790:446c46af4b42 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 | 2645f9ef8c88 |
| rev | line source |
|---|---|
|
17744
d63878346099
maint: Update copyright notices for release.
John W. Eaton <jwe@octave.org>
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1 ## Copyright (C) 2006, 2013 Sylvain Pelissier |
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2 ## |
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3 ## This file is part of Octave. |
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4 ## |
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5 ## Octave is free software; you can redistribute it and/or modify it |
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6 ## under the terms of the GNU General Public License as published by |
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7 ## the Free Software Foundation; either version 3 of the License, or (at |
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8 ## your option) any later version. |
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9 ## |
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10 ## Octave is distributed in the hope that it will be useful, but |
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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13 ## General Public License for more details. |
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14 ## |
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15 ## You should have received a copy of the GNU General Public License |
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16 ## along with Octave; see the file COPYING. If not, see |
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17 ## <http://www.gnu.org/licenses/>. |
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18 |
| 16768 | 19 ## Author: Sylvain Pelissier <sylvain.pelissier@gmail.com> |
| 20 | |
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21 ## -*- texinfo -*- |
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22 ## @deftypefn {Function File} {} expint (@var{x}) |
| 17363 | 23 ## Compute the exponential integral: |
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24 ## @tex |
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25 ## $$ |
| 17363 | 26 ## {\rm E_1} (x) = \int_x^\infty {e^{-t} \over t} dt |
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27 ## $$ |
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28 ## @end tex |
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29 ## @ifnottex |
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30 ## |
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31 ## @example |
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32 ## @group |
| 17363 | 33 ## infinity |
| 34 ## / | |
| 35 ## E_1 (x) = | exp (-t)/t dt | |
| 36 ## / | |
| 37 ## x | |
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38 ## @end group |
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39 ## @end example |
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40 ## |
| 17363 | 41 ## @end ifnottex |
| 42 ## Note: For compatibility, this functions uses the @sc{matlab} definition | |
| 43 ## of the exponential integral. Most other sources refer to this particular | |
| 44 ## value as @math{E_1 (x)}, and the exponential integral is | |
| 45 ## @tex | |
| 46 ## $$ | |
| 47 ## {\rm Ei} (x) = - \int_{-x}^\infty {e^{-t} \over t} dt. | |
| 48 ## $$ | |
| 49 ## @end tex | |
| 50 ## @ifnottex | |
| 51 ## | |
| 52 ## @example | |
| 53 ## @group | |
| 54 ## infinity | |
| 55 ## / | |
| 56 ## Ei (x) = - | exp (-t)/t dt | |
| 57 ## / | |
| 58 ## -x | |
| 59 ## @end group | |
| 60 ## @end example | |
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61 ## |
| 17363 | 62 ## @end ifnottex |
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63 ## The two definitions are related, for positive real values of @var{x}, by |
| 17363 | 64 ## @tex |
| 65 ## $ | |
| 66 ## E_1 (-x) = -{\rm Ei} (x) - i\pi. | |
| 67 ## $ | |
| 68 ## @end tex | |
| 69 ## @ifnottex | |
| 70 ## @w{@code{E_1 (-x) = -Ei (x) - i*pi}}. | |
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71 ## @end ifnottex |
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72 ## @end deftypefn |
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73 |
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74 function y = expint (x) |
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75 |
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76 if (nargin != 1) |
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77 print_usage (); |
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78 endif |
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79 |
| 17363 | 80 y = x; # Copy over all values, including NaNs |
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81 |
| 17363 | 82 if (isreal (x)) |
| 83 idx = (x >= 0); | |
| 84 y(idx) = -expint_Ei (-x(idx)); | |
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85 |
| 17363 | 86 idx = (x < 0); |
| 87 y(idx) = -expint_Ei (-x(idx)) - i*pi; | |
| 88 else | |
| 89 idx = (imag (x) > 0); | |
| 90 y(idx) = -expint_Ei (-x(idx)) - i*pi; | |
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91 |
| 17363 | 92 idx = (imag (x) < 0); |
| 93 y(idx) = -expint_Ei (-x(idx)) + i*pi; | |
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94 |
| 17363 | 95 isreal_idx = (imag (x) == 0); |
| 96 idx = (isreal_idx & real (x) >= 0); | |
| 97 y(idx) = -expint_Ei (-x(idx)); | |
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98 |
| 17363 | 99 idx = (isreal_idx & real (x) < 0); |
| 100 y(idx) = -expint_Ei (-x(idx)) - i*pi; | |
| 101 endif | |
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102 |
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103 endfunction |
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104 |
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105 ## -*- texinfo -*- |
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106 ## @deftypefn {Function File} {@var{y} =} expint_Ei (@var{x}) |
| 17363 | 107 ## Compute the exponential integral: |
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108 ## @verbatim |
| 17363 | 109 ## infinity |
| 110 ## / | |
| 111 ## expint_Ei (x) = - | exp(-t)/t dt | |
| 112 ## / | |
| 113 ## -x | |
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114 ## @end verbatim |
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115 ## @end deftypefn |
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116 |
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117 function y = expint_Ei (x) |
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118 |
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119 y = zeros (size (x)); |
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120 F = @(x) exp (-x)./x; |
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121 |
| 17363 | 122 for t = 1:numel (x) |
| 123 xt = x(t); | |
| 124 if (xt < 0 && imag (xt) == 0) | |
| 125 ## Direct integration for most real inputs | |
| 126 y(t) = -quad (F, -xt, Inf, [0, 1e-10]); | |
| 127 elseif (xt > 2 && imag (xt) == 0) | |
| 128 persistent Ei_2 = 4.954234356001890; | |
| 129 y(t) = Ei_2 - quad (F, -xt, -2); | |
| 130 elseif (abs (xt) < 10) | |
| 131 ## Series Expansion for real (range [0,2]) or complex inputs (r < 10) | |
| 132 k = 1; | |
| 133 do | |
| 134 term = xt^k / (k*factorial (k)); | |
| 135 y(t) += term; | |
| 136 until (abs (term) < eps (abs (y(t))) / 2 || k++ >= 100) | |
| 137 y(t) = 0.57721566490153286 + log (xt) + y(t); | |
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138 else |
| 17363 | 139 ## FIXME: This expansion is accurate to only 1e-13 at the beginning |
| 140 ## near 10+i, although it becomes more accurate as the magnitude | |
| 141 ## of xt grows. | |
| 142 if (imag (xt) <= 0) | |
| 143 persistent a1 = 4.03640; | |
| 144 persistent a2 = 1.15198; | |
| 145 persistent b1 = 5.03637; | |
| 146 persistent b2 = 4.19160; | |
| 147 y(t) = -(xt^2 - a1*xt + a2) ... | |
| 148 / ((xt^2 - b1*xt + b2) * (-xt) * exp (-xt)) ... | |
| 149 - i*pi; | |
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150 else |
| 17363 | 151 y(t) = conj (expint_Ei (conj (xt))); |
| 152 endif; | |
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153 endif |
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154 endfor |
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155 endfunction |
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156 |
| 17363 | 157 |
| 158 ## Test against A&S Table 5.1 | |
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159 %!test |
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160 %! x = [5:5:50]'/100; |
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161 %! gamma = 0.5772156649; |
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162 %! y_exp = [0.9876375971; |
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163 %! 0.9755453033; |
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164 %! 0.9637156702; |
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165 %! 0.9521414833; |
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166 %! 0.9408157528; |
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167 %! 0.9297317075; |
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168 %! 0.9188827858; |
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169 %! 0.9082626297; |
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170 %! 0.8978650778; |
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171 %! 0.8876841584 ]; |
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172 %! y = (expint (x) + log(x) + gamma) ./ x; |
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173 %! assert (y, y_exp, 1e-9); |
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174 %!test |
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175 %! x = [50:5:95]'/100; |
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176 %! y_exp = [0.559773595; |
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177 %! 0.503364081; |
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178 %! 0.454379503; |
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179 %! 0.411516976; |
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180 %! 0.373768843; |
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181 %! 0.340340813; |
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182 %! 0.310596579; |
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183 %! 0.284019269; |
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184 %! 0.260183939; |
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185 %! 0.238737524 ]; |
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186 %! y = expint (x); |
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187 %! assert (y, y_exp, 1e-9); |
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188 %!test |
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189 %! x = [100:5:145]'/100; |
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190 %! y_exp = [0.219383934; |
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191 %! 0.201872813; |
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192 %! 0.185990905; |
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193 %! 0.171555354; |
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194 %! 0.158408437; |
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195 %! 0.146413373; |
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196 %! 0.135450958; |
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197 %! 0.125416844; |
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198 %! 0.116219313; |
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199 %! 0.107777440 ]; |
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200 %! y = expint (x); |
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201 %! assert (y, y_exp, 1e-9); |
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202 %!test |
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203 %! x = [150:5:200]'/100; |
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204 %! y_exp = [0.100019582; |
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205 %! 0.092882108; |
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206 %! 0.086308334; |
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207 %! 0.080247627; |
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208 %! 0.074654644; |
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209 %! 0.069488685; |
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210 %! 0.064713129; |
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211 %! 0.060294967; |
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212 %! 0.056204378; |
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213 %! 0.052414380; |
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214 %! 0.048900511 ]; |
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215 %! y = expint (x); |
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216 %! assert (y, y_exp, 1e-9); |
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217 |
| 17363 | 218 ## Series expansion (-2 < x < 0) |
| 219 ## Expected values from Mathematica | |
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220 %!test |
| 17363 | 221 %! x = [-0.1; -0.5; -1; -1.5; -2]; |
| 222 %! y_exp = [ 1.6228128139692767 - i*pi; | |
| 223 %! -0.45421990486317358 - i*pi; | |
| 224 %! -1.8951178163559368 - i*pi; | |
| 225 %! -3.3012854491297978 - i*pi; | |
| 226 %! -4.9542343560018902 - i*pi]; | |
| 227 %! y = expint (x); | |
| 228 %! assert (y, y_exp, eps (real (y_exp))); | |
| 229 | |
| 230 ## (x < -2, x real) | |
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231 %!test |
| 17363 | 232 %! x = [-2.5; -3; -10;-15; -25]; |
| 233 %! y_exp = [-7.0737658945786007 - i*pi; | |
| 234 %! -9.9338325706254165 - i*pi; | |
| 235 %! -2492.2289762418777 - i*pi; | |
| 236 %! -234955.85249076830 - i*pi; | |
| 237 %! -3.0059509065255486e9 - i*pi]; | |
| 238 %! y = expint (x); | |
| 239 %! assert (y, y_exp, 8*eps (real (y_exp))); | |
| 240 | |
| 241 ## Complex values | |
| 242 %!test | |
| 243 %! x = [i; -1-i; 10-i; 10+i]; | |
| 244 %! y_exp = [-0.33740392290096813 - i*0.62471325642771360; | |
| 245 %! -1.7646259855638540 + i*0.75382280207927082; | |
| 246 %! 1.90746381979783120e-6 + i*3.67354374003294739e-6; | |
| 247 %! 1.90746381979783120e-6 - i*3.67354374003294739e-6]; | |
| 248 %! y = expint (x); | |
| 249 %! assert (y, y_exp, 1e-12); | |
| 250 | |
| 251 ## Exceptional values (-Inf, Inf, NaN, 0, 0.37250741078) | |
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252 %!test |
| 17363 | 253 %! x = [-Inf; Inf; NaN; 0; -0.3725074107813668]; |
| 254 %! y_exp = [-Inf - i*pi; | |
| 255 %! -Inf; # should be 0; | |
| 256 %! NaN; | |
| 257 %! Inf; | |
| 258 %! 0 - i*pi]; | |
| 259 %! y = expint (x); | |
| 260 %! assert (y, y_exp, 5*eps); | |
| 261 | |
| 262 ## Test input validation | |
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263 %!error expint () |
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264 %!error expint (1,2) |
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265 |
| 17363 | 266 |