Mercurial > octave-nkf
view scripts/specfun/legendre.m @ 17281:bc924baa2c4e
doc: Add new @qcode macro for code samples which are quoted.
Macro handles options ("on") or properties ("position") more elegantly
than @code{"text"}.
* doc/interpreter/macros.texi: Add new @qcode macro.
* doc/interpreter/tips.txi: Add documentation about @qcode macro.
* doc/interpreter/basics.txi, doc/interpreter/container.txi,
doc/interpreter/emacs.txi, doc/interpreter/errors.txi,
doc/interpreter/eval.txi, doc/interpreter/expr.txi,
doc/interpreter/external.txi, doc/interpreter/func.txi,
doc/interpreter/grammar.txi, doc/interpreter/image.txi,
doc/interpreter/install.txi, doc/interpreter/interp.txi,
doc/interpreter/io.txi, doc/interpreter/matrix.txi,
doc/interpreter/numbers.txi, doc/interpreter/oop.txi,
doc/interpreter/package.txi, doc/interpreter/plot.txi,
doc/interpreter/quad.txi, doc/interpreter/sparse.txi,
doc/interpreter/strings.txi, doc/interpreter/system.txi,
doc/interpreter/vectorize.txi, libinterp/corefcn/balance.cc,
libinterp/corefcn/bitfcns.cc, libinterp/corefcn/cellfun.cc,
libinterp/corefcn/conv2.cc, libinterp/corefcn/data.cc,
libinterp/corefcn/debug.cc, libinterp/corefcn/defaults.cc,
libinterp/corefcn/dirfns.cc, libinterp/corefcn/dlmread.cc,
libinterp/corefcn/error.cc, libinterp/corefcn/file-io.cc,
libinterp/corefcn/find.cc, libinterp/corefcn/gammainc.cc,
libinterp/corefcn/graphics.cc, libinterp/corefcn/help.cc,
libinterp/corefcn/hex2num.cc, libinterp/corefcn/input.cc,
libinterp/corefcn/load-path.cc, libinterp/corefcn/load-save.cc,
libinterp/corefcn/ls-oct-ascii.cc, libinterp/corefcn/lu.cc,
libinterp/corefcn/luinc.cc, libinterp/corefcn/matrix_type.cc,
libinterp/corefcn/oct-hist.cc, libinterp/corefcn/pager.cc,
libinterp/corefcn/pr-output.cc, libinterp/corefcn/pt-jit.cc,
libinterp/corefcn/qz.cc, libinterp/corefcn/rand.cc,
libinterp/corefcn/regexp.cc, libinterp/corefcn/schur.cc,
libinterp/corefcn/sighandlers.cc, libinterp/corefcn/sparse.cc,
libinterp/corefcn/spparms.cc, libinterp/corefcn/str2double.cc,
libinterp/corefcn/svd.cc, libinterp/corefcn/symtab.cc,
libinterp/corefcn/syscalls.cc, libinterp/corefcn/toplev.cc,
libinterp/corefcn/tril.cc, libinterp/corefcn/typecast.cc,
libinterp/corefcn/utils.cc, libinterp/corefcn/variables.cc,
libinterp/dldfcn/__init_fltk__.cc, libinterp/dldfcn/chol.cc,
libinterp/dldfcn/colamd.cc, libinterp/dldfcn/fftw.cc, libinterp/dldfcn/qr.cc,
libinterp/dldfcn/symbfact.cc, libinterp/octave-value/ov-base.cc,
libinterp/octave-value/ov-fcn-handle.cc,
libinterp/octave-value/ov-fcn-inline.cc, libinterp/octave-value/ov-java.cc,
libinterp/octave-value/ov-range.cc, libinterp/octave-value/ov-struct.cc,
libinterp/octave-value/ov-usr-fcn.cc, libinterp/parse-tree/oct-parse.in.yy,
libinterp/parse-tree/pt-binop.cc, libinterp/parse-tree/pt-eval.cc,
libinterp/parse-tree/pt-mat.cc, scripts/@ftp/ftp.m,
scripts/deprecated/java_convert_matrix.m, scripts/deprecated/java_debug.m,
scripts/deprecated/java_unsigned_conversion.m, scripts/deprecated/shell_cmd.m,
scripts/general/dblquad.m, scripts/general/display.m,
scripts/general/genvarname.m, scripts/general/idivide.m,
scripts/general/interp1.m, scripts/general/interp2.m,
scripts/general/interp3.m, scripts/general/interpn.m, scripts/general/isa.m,
scripts/general/profexplore.m, scripts/general/profile.m,
scripts/general/quadgk.m, scripts/general/randi.m, scripts/general/structfun.m,
scripts/general/subsindex.m, scripts/general/triplequad.m,
scripts/geometry/griddata.m, scripts/geometry/griddata3.m,
scripts/geometry/griddatan.m, scripts/geometry/voronoi.m, scripts/help/help.m,
scripts/help/lookfor.m, scripts/image/cmpermute.m, scripts/image/colormap.m,
scripts/image/image.m, scripts/image/imagesc.m, scripts/image/imfinfo.m,
scripts/image/imformats.m, scripts/image/imread.m, scripts/image/imshow.m,
scripts/image/imwrite.m, scripts/image/ind2gray.m, scripts/image/lines.m,
scripts/image/rgb2ind.m, scripts/image/spinmap.m, scripts/io/dlmwrite.m,
scripts/io/strread.m, scripts/io/textread.m, scripts/io/textscan.m,
scripts/java/javaclasspath.m, scripts/java/usejava.m,
scripts/miscellaneous/bzip2.m, scripts/miscellaneous/computer.m,
scripts/miscellaneous/copyfile.m, scripts/miscellaneous/debug.m,
scripts/miscellaneous/dos.m, scripts/miscellaneous/edit.m,
scripts/miscellaneous/gzip.m, scripts/miscellaneous/license.m,
scripts/miscellaneous/mkoctfile.m, scripts/miscellaneous/movefile.m,
scripts/miscellaneous/parseparams.m, scripts/miscellaneous/unix.m,
scripts/optimization/fminbnd.m, scripts/optimization/fminsearch.m,
scripts/optimization/fminunc.m, scripts/optimization/fsolve.m,
scripts/optimization/fzero.m, scripts/optimization/glpk.m,
scripts/optimization/lsqnonneg.m, scripts/optimization/optimset.m,
scripts/optimization/pqpnonneg.m, scripts/pkg/pkg.m, scripts/plot/allchild.m,
scripts/plot/ancestor.m, scripts/plot/area.m, scripts/plot/axis.m,
scripts/plot/bar.m, scripts/plot/barh.m, scripts/plot/box.m,
scripts/plot/caxis.m, scripts/plot/cla.m, scripts/plot/clabel.m,
scripts/plot/clf.m, scripts/plot/close.m, scripts/plot/colorbar.m,
scripts/plot/daspect.m, scripts/plot/ezmesh.m, scripts/plot/ezmeshc.m,
scripts/plot/ezsurf.m, scripts/plot/ezsurfc.m, scripts/plot/findall.m,
scripts/plot/findobj.m, scripts/plot/gcbo.m, scripts/plot/gcf.m,
scripts/plot/gco.m, scripts/plot/grid.m, scripts/plot/guihandles.m,
scripts/plot/hdl2struct.m, scripts/plot/hidden.m, scripts/plot/hold.m,
scripts/plot/isonormals.m, scripts/plot/isosurface.m, scripts/plot/legend.m,
scripts/plot/mesh.m, scripts/plot/meshc.m, scripts/plot/meshz.m,
scripts/plot/newplot.m, scripts/plot/orient.m, scripts/plot/pareto.m,
scripts/plot/patch.m, scripts/plot/pbaspect.m, scripts/plot/pcolor.m,
scripts/plot/plot.m, scripts/plot/print.m,
scripts/plot/private/__add_default_menu__.m, scripts/plot/quiver.m,
scripts/plot/quiver3.m, scripts/plot/refreshdata.m, scripts/plot/saveas.m,
scripts/plot/scatter.m, scripts/plot/scatter3.m, scripts/plot/shading.m,
scripts/plot/shrinkfaces.m, scripts/plot/slice.m, scripts/plot/stem.m,
scripts/plot/stem3.m, scripts/plot/struct2hdl.m, scripts/plot/subplot.m,
scripts/plot/surf.m, scripts/plot/surfc.m, scripts/plot/surfl.m,
scripts/plot/tetramesh.m, scripts/plot/uigetfile.m, scripts/plot/uimenu.m,
scripts/plot/uiputfile.m, scripts/plot/waterfall.m, scripts/plot/whitebg.m,
scripts/plot/xlim.m, scripts/plot/ylim.m, scripts/plot/zlim.m,
scripts/polynomial/conv.m, scripts/polynomial/polyout.m,
scripts/polynomial/splinefit.m, scripts/set/ismember.m, scripts/set/powerset.m,
scripts/set/setdiff.m, scripts/set/union.m, scripts/set/unique.m,
scripts/signal/detrend.m, scripts/signal/filter2.m, scripts/signal/freqz.m,
scripts/signal/periodogram.m, scripts/signal/spectral_adf.m,
scripts/signal/spectral_xdf.m, scripts/sparse/eigs.m, scripts/sparse/svds.m,
scripts/specfun/legendre.m, scripts/special-matrix/gallery.m,
scripts/statistics/base/mean.m, scripts/statistics/base/moment.m,
scripts/statistics/tests/cor_test.m,
scripts/statistics/tests/kolmogorov_smirnov_test.m,
scripts/statistics/tests/kolmogorov_smirnov_test_2.m,
scripts/statistics/tests/kruskal_wallis_test.m,
scripts/statistics/tests/prop_test_2.m, scripts/statistics/tests/sign_test.m,
scripts/statistics/tests/t_test.m, scripts/statistics/tests/t_test_2.m,
scripts/statistics/tests/t_test_regression.m,
scripts/statistics/tests/u_test.m, scripts/statistics/tests/var_test.m,
scripts/statistics/tests/welch_test.m,
scripts/statistics/tests/wilcoxon_test.m, scripts/statistics/tests/z_test.m,
scripts/statistics/tests/z_test_2.m, scripts/strings/base2dec.m,
scripts/strings/index.m, scripts/strings/isstrprop.m,
scripts/strings/mat2str.m, scripts/strings/regexptranslate.m,
scripts/strings/rindex.m, scripts/strings/str2num.m, scripts/strings/strcat.m,
scripts/strings/strjust.m, scripts/strings/strmatch.m,
scripts/strings/validatestring.m, scripts/testfun/demo.m,
scripts/testfun/example.m, scripts/testfun/test.m, scripts/time/addtodate.m,
scripts/time/asctime.m, scripts/time/datestr.m, scripts/time/datetick.m,
scripts/time/weekday.m, scripts/ui/errordlg.m, scripts/ui/helpdlg.m,
scripts/ui/inputdlg.m, scripts/ui/listdlg.m, scripts/ui/msgbox.m,
scripts/ui/questdlg.m, scripts/ui/warndlg.m: Use new @qcode macro.
author | Rik <rik@octave.org> |
---|---|
date | Mon, 19 Aug 2013 20:46:38 -0700 |
parents | 5d3a684236b0 |
children | d63878346099 |
line wrap: on
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## Copyright (C) 2000-2012 Kai Habel ## Copyright (C) 2008 Marco Caliari ## ## 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{l} =} legendre (@var{n}, @var{x}) ## @deftypefnx {Function File} {@var{l} =} legendre (@var{n}, @var{x}, @var{normalization}) ## Compute the Legendre function of degree @var{n} and order ## @var{m} = 0 @dots{} N@. The optional argument, @var{normalization}, ## may be one of @qcode{"unnorm"}, @qcode{"sch"}, or @qcode{"norm"}. ## The default is @qcode{"unnorm"}. The value of @var{n} must be a ## non-negative scalar integer. ## ## If the optional argument @var{normalization} is missing or is ## @qcode{"unnorm"}, compute the Legendre function of degree @var{n} and ## order @var{m} and return all values for @var{m} = 0 @dots{} @var{n}. ## The return value has one dimension more than @var{x}. ## ## The Legendre Function of degree @var{n} and order @var{m}: ## ## @tex ## $$ ## P^m_n(x) = (-1)^m (1-x^2)^{m/2}{d^m\over {dx^m}}P_n (x) ## $$ ## @end tex ## @ifnottex ## ## @example ## @group ## m m 2 m/2 d^m ## P(x) = (-1) * (1-x ) * ---- P(x) ## n dx^m n ## @end group ## @end example ## ## @end ifnottex ## ## @noindent ## with Legendre polynomial of degree @var{n}: ## ## @tex ## $$ ## P(x) = {1\over{2^n n!}}\biggl({d^n\over{dx^n}}(x^2 - 1)^n\biggr) ## $$ ## @end tex ## @ifnottex ## ## @example ## @group ## 1 d^n 2 n ## P(x) = ------ [----(x - 1) ] ## n 2^n n! dx^n ## @end group ## @end example ## ## @end ifnottex ## ## @noindent ## @code{legendre (3, [-1.0, -0.9, -0.8])} returns the matrix: ## ## @example ## @group ## x | -1.0 | -0.9 | -0.8 ## ------------------------------------ ## m=0 | -1.00000 | -0.47250 | -0.08000 ## m=1 | 0.00000 | -1.99420 | -1.98000 ## m=2 | 0.00000 | -2.56500 | -4.32000 ## m=3 | 0.00000 | -1.24229 | -3.24000 ## @end group ## @end example ## ## If the optional argument @code{normalization} is @qcode{"sch"}, ## compute the Schmidt semi-normalized associated Legendre function. ## The Schmidt semi-normalized associated Legendre function is related ## to the unnormalized Legendre functions by the following: ## ## For Legendre functions of degree n and order 0: ## ## @tex ## $$ ## SP^0_n (x) = P^0_n (x) ## $$ ## @end tex ## @ifnottex ## ## @example ## @group ## 0 0 ## SP(x) = P(x) ## n n ## @end group ## @end example ## ## @end ifnottex ## ## For Legendre functions of degree n and order m: ## ## @tex ## $$ ## SP^m_n (x) = P^m_n (x)(-1)^m\biggl({2(n-m)!\over{(n+m)!}}\biggl)^{0.5} ## $$ ## @end tex ## @ifnottex ## ## @example ## @group ## m m m 2(n-m)! 0.5 ## SP(x) = P(x) * (-1) * [-------] ## n n (n+m)! ## @end group ## @end example ## ## @end ifnottex ## ## If the optional argument @var{normalization} is @qcode{"norm"}, ## compute the fully normalized associated Legendre function. ## The fully normalized associated Legendre function is related ## to the unnormalized Legendre functions by the following: ## ## For Legendre functions of degree @var{n} and order @var{m} ## ## @tex ## $$ ## NP^m_n (x) = P^m_n (x)(-1)^m\biggl({(n+0.5)(n-m)!\over{(n+m)!}}\biggl)^{0.5} ## $$ ## @end tex ## @ifnottex ## ## @example ## @group ## m m m (n+0.5)(n-m)! 0.5 ## NP(x) = P(x) * (-1) * [-------------] ## n n (n+m)! ## @end group ## @end example ## ## @end ifnottex ## @end deftypefn ## Author: Marco Caliari <marco.caliari@univr.it> function retval = legendre (n, x, normalization) persistent warned_overflow = false; if (nargin < 2 || nargin > 3) print_usage (); endif if (!isscalar (n) || n < 0 || n != fix (n)) error ("legendre: N must be a non-negative scalar integer"); endif if (!isreal (x) || any (x(:) < -1 | x(:) > 1)) error ("legendre: X must be real-valued vector in the range -1 <= X <= 1"); endif if (nargin == 3) normalization = lower (normalization); else normalization = "unnorm"; endif switch (normalization) case "norm" scale = sqrt (n+0.5); case "sch" scale = sqrt (2); case "unnorm" scale = 1; otherwise error ('legendre: expecting NORMALIZATION option to be "norm", "sch", or "unnorm"'); endswitch scale = scale * ones (size (x)); ## Based on the recurrence relation below ## m m m ## (n-m+1) * P (x) = (2*n+1)*x*P (x) - (n+1)*P (x) ## n+1 n n-1 ## http://en.wikipedia.org/wiki/Associated_Legendre_function overflow = false; retval = zeros ([n+1, size(x)]); for m = 1:n lpm1 = scale; lpm2 = (2*m-1) .* x .* scale; lpm3 = lpm2; for k = m+1:n lpm3a = (2*k-1) .* x .* lpm2; lpm3b = (k+m-2) .* lpm1; lpm3 = (lpm3a - lpm3b)/(k-m+1); lpm1 = lpm2; lpm2 = lpm3; if (! warned_overflow) if (any (abs (lpm3a) > realmax) || any (abs (lpm3b) > realmax) || any (abs (lpm3) > realmax)) overflow = true; endif endif endfor retval(m,:) = lpm3(:); if (strcmp (normalization, "unnorm")) scale = -scale * (2*m-1); else ## normalization == "sch" or normalization == "norm" scale = scale / sqrt ((n-m+1)*(n+m))*(2*m-1); endif scale = scale .* sqrt (1-x.^2); endfor retval(n+1,:) = scale(:); if (isvector (x)) ## vector case is special retval = reshape (retval, n + 1, length (x)); endif if (strcmp (normalization, "sch")) retval(1,:) = retval(1,:) / sqrt (2); endif if (overflow && ! warned_overflow) warning ("legendre: overflow - results may be unstable for high orders"); warned_overflow = true; endif endfunction %!test %! result = legendre (3, [-1.0 -0.9 -0.8]); %! expected = [ %! -1.00000 -0.47250 -0.08000 %! 0.00000 -1.99420 -1.98000 %! 0.00000 -2.56500 -4.32000 %! 0.00000 -1.24229 -3.24000 %! ]; %! assert (result, expected, 1e-5); %!test %! result = legendre (3, [-1.0 -0.9 -0.8], "sch"); %! expected = [ %! -1.00000 -0.47250 -0.08000 %! 0.00000 0.81413 0.80833 %! -0.00000 -0.33114 -0.55771 %! 0.00000 0.06547 0.17076 %! ]; %! assert (result, expected, 1e-5); %!test %! result = legendre (3, [-1.0 -0.9 -0.8], "norm"); %! expected = [ %! -1.87083 -0.88397 -0.14967 %! 0.00000 1.07699 1.06932 %! -0.00000 -0.43806 -0.73778 %! 0.00000 0.08661 0.22590 %! ]; %! assert (result, expected, 1e-5); %!test %! result = legendre (151, 0); %! ## Don't compare to "-Inf" since it would fail on 64 bit systems. %! assert (result(end) < -1.7976e308 && all (isfinite (result(1:end-1)))); %!test %! result = legendre (150, 0); %! ## This agrees with Matlab's result. %! assert (result(end), 3.7532741115719e+306, 0.0000000000001e+306); %!test %! result = legendre (0, 0:0.1:1); %! assert (result, full (ones (1,11))); %!test %! result = legendre (3, [-1,0,1;1,0,-1]); %! ## Test matrix input %! expected(:,:,1) = [-1,1;0,0;0,0;0,0]; %! expected(:,:,2) = [0,0;1.5,1.5;0,0;-15,-15]; %! expected(:,:,3) = [1,-1;0,0;0,0;0,0]; %! assert (result, expected); %!test %! result = legendre (3, [-1,0,1;1,0,-1]'); %! expected(:,:,1) = [-1,0,1;0,1.5,0;0,0,0;0,-15,0]; %! expected(:,:,2) = [1,0,-1;0,1.5,0;0,0,0;0,-15,0]; %! assert (result, expected); %% Check correct invocation %!error legendre () %!error legendre (1) %!error legendre (1,2,3,4) %!error legendre ([1, 2], [-1, 0, 1]) %!error legendre (-1, [-1, 0, 1]) %!error legendre (1.1, [-1, 0, 1]) %!error legendre (1, [-1+i, 0, 1]) %!error legendre (1, [-2, 0, 1]) %!error legendre (1, [-1, 0, 2]) %!error legendre (1, [-1, 0, 1], "badnorm")