Mercurial > octave-libtiff
annotate scripts/special-matrix/hilb.m @ 30875:5d3faba0342e
doc: Ensure documentation lists output argument when it exists for all m-files.
For new users of Octave it is best to show explicit calling forms
in the documentation and to show a return argument when it exists.
* bp-table.cc, shift.m, accumarray.m, accumdim.m, bincoeff.m, bitcmp.m,
bitget.m, bitset.m, blkdiag.m, celldisp.m, cplxpair.m, dblquad.m, flip.m,
fliplr.m, flipud.m, idivide.m, int2str.m, interpft.m, logspace.m, num2str.m,
polyarea.m, postpad.m, prepad.m, randi.m, repmat.m, rng.m, rot90.m, rotdim.m,
structfun.m, triplequad.m, uibuttongroup.m, uicontrol.m, uipanel.m,
uipushtool.m, uitoggletool.m, uitoolbar.m, waitforbuttonpress.m, help.m,
__additional_help_message__.m, hsv.m, im2double.m, im2frame.m, javachk.m,
usejava.m, argnames.m, char.m, formula.m, inline.m, __vectorize__.m, findstr.m,
flipdim.m, strmatch.m, vectorize.m, commutation_matrix.m, cond.m, cross.m,
duplication_matrix.m, expm.m, orth.m, rank.m, rref.m, trace.m, vech.m, cast.m,
compare_versions.m, delete.m, dir.m, fileattrib.m, grabcode.m, gunzip.m,
inputname.m, license.m, list_primes.m, ls.m, mexext.m, movefile.m,
namelengthmax.m, nargoutchk.m, nthargout.m, substruct.m, swapbytes.m, ver.m,
verLessThan.m, what.m, fminunc.m, fsolve.m, fzero.m, optimget.m, __fdjac__.m,
matlabroot.m, savepath.m, campos.m, camroll.m, camtarget.m, camup.m, camva.m,
camzoom.m, clabel.m, diffuse.m, legend.m, orient.m, rticks.m, specular.m,
thetaticks.m, xlim.m, xtickangle.m, xticklabels.m, xticks.m, ylim.m,
ytickangle.m, yticklabels.m, yticks.m, zlim.m, ztickangle.m, zticklabels.m,
zticks.m, ellipsoid.m, isocolors.m, isonormals.m, stairs.m, surfnorm.m,
__actual_axis_position__.m, __pltopt__.m, close.m, graphics_toolkit.m, pan.m,
print.m, printd.m, __ghostscript__.m, __gnuplot_print__.m, __opengl_print__.m,
rotate3d.m, subplot.m, zoom.m, compan.m, conv.m, poly.m, polyaffine.m,
polyder.m, polyint.m, polyout.m, polyreduce.m, polyvalm.m, roots.m, prefdir.m,
prefsfile.m, profexplore.m, profexport.m, profshow.m, powerset.m, unique.m,
arch_rnd.m, arma_rnd.m, autoreg_matrix.m, bartlett.m, blackman.m, detrend.m,
durbinlevinson.m, fftconv.m, fftfilt.m, fftshift.m, fractdiff.m, hamming.m,
hanning.m, hurst.m, ifftshift.m, rectangle_lw.m, rectangle_sw.m, triangle_lw.m,
sinc.m, sinetone.m, sinewave.m, spectral_adf.m, spectral_xdf.m, spencer.m,
ilu.m, __sprand__.m, sprand.m, sprandn.m, sprandsym.m, treelayout.m, beta.m,
betainc.m, betaincinv.m, betaln.m, cosint.m, expint.m, factorial.m, gammainc.m,
gammaincinv.m, lcm.m, nthroot.m, perms.m, reallog.m, realpow.m, realsqrt.m,
sinint.m, hadamard.m, hankel.m, hilb.m, invhilb.m, magic.m, pascal.m, rosser.m,
toeplitz.m, vander.m, wilkinson.m, center.m, corr.m, cov.m, discrete_cdf.m,
discrete_inv.m, discrete_pdf.m, discrete_rnd.m, empirical_cdf.m,
empirical_inv.m, empirical_pdf.m, empirical_rnd.m, kendall.m, kurtosis.m,
mad.m, mean.m, meansq.m, median.m, mode.m, moment.m, range.m, ranks.m,
run_count.m, skewness.m, spearman.m, statistics.m, std.m, base2dec.m,
bin2dec.m, blanks.m, cstrcat.m, deblank.m, dec2base.m, dec2bin.m, dec2hex.m,
hex2dec.m, index.m, regexptranslate.m, rindex.m, strcat.m, strjust.m,
strtrim.m, strtrunc.m, substr.m, untabify.m, __have_feature__.m,
__prog_output_assert__.m, __run_test_suite__.m, example.m, fail.m, asctime.m,
calendar.m, ctime.m, date.m, etime.m:
Add return arguments to @deftypefn macros where they were missing. Rename
variables in functions (particularly generic "retval") to match documentation.
Rename some return variables for (hopefully) better clarity (e.g., 'ax' to 'hax'
to indicate it is a graphics handle to an axes object).
author | Rik <rik@octave.org> |
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date | Wed, 30 Mar 2022 20:40:27 -0700 |
parents | 796f54d4ddbf |
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rev | line source |
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1 ######################################################################## |
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2 ## |
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3 ## Copyright (C) 1993-2022 The Octave Project Developers |
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4 ## |
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5 ## See the file COPYRIGHT.md in the top-level directory of this |
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6 ## distribution or <https://octave.org/copyright/>. |
2313 | 7 ## |
8 ## This file is part of Octave. | |
9 ## | |
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10 ## Octave is free software: you can redistribute it and/or modify it |
2313 | 11 ## under the terms of the GNU General Public License as published by |
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12 ## the Free Software Foundation, either version 3 of the License, or |
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13 ## (at your option) any later version. |
2313 | 14 ## |
15 ## Octave is distributed in the hope that it will be useful, but | |
16 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
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17 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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18 ## GNU General Public License for more details. |
2313 | 19 ## |
20 ## You should have received a copy of the GNU General Public License | |
7016 | 21 ## along with Octave; see the file COPYING. If not, see |
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22 ## <https://www.gnu.org/licenses/>. |
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23 ## |
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24 ######################################################################## |
245 | 25 |
3369 | 26 ## -*- texinfo -*- |
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27 ## @deftypefn {} {@var{h} =} hilb (@var{n}) |
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28 ## Return the Hilbert matrix of order @var{n}. |
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29 ## |
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30 ## The @math{i,j} element of a Hilbert matrix is defined as |
3369 | 31 ## @tex |
32 ## $$ | |
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33 ## H(i, j) = {1 \over (i + j - 1)} |
3369 | 34 ## $$ |
35 ## @end tex | |
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36 ## @ifnottex |
3426 | 37 ## |
3369 | 38 ## @example |
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39 ## H(i, j) = 1 / (i + j - 1) |
3369 | 40 ## @end example |
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41 ## |
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42 ## @end ifnottex |
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43 ## |
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44 ## Hilbert matrices are close to being singular which make them difficult to |
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45 ## invert with numerical routines. Comparing the condition number of a random |
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46 ## matrix 5x5 matrix with that of a Hilbert matrix of order 5 reveals just how |
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47 ## difficult the problem is. |
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48 ## |
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49 ## @example |
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50 ## @group |
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51 ## cond (rand (5)) |
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52 ## @result{} 14.392 |
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53 ## cond (hilb (5)) |
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54 ## @result{} 4.7661e+05 |
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55 ## @end group |
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56 ## @end example |
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57 ## |
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58 ## @seealso{invhilb} |
3369 | 59 ## @end deftypefn |
4 | 60 |
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61 function h = hilb (n) |
4 | 62 |
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63 if (nargin != 1) |
6046 | 64 print_usage (); |
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65 elseif (! isscalar (n)) |
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66 error ("hilb: N must be a scalar integer"); |
4 | 67 endif |
68 | |
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69 ## Very elegant solution by N. Higham |
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70 ## https://nhigham.com/2020/06/30/what-is-the-hilbert-matrix/ |
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71 j = 1:n; |
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72 h = 1 ./ (j' + j - 1); |
4 | 73 |
74 endfunction | |
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76 |
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77 %!assert (hilb (2), [1, 1/2; 1/2, 1/3]) |
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78 %!assert (hilb (3), [1, 1/2, 1/3; 1/2, 1/3, 1/4; 1/3, 1/4, 1/5]) |
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80 %!error <Invalid call> hilb () |
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81 %!error <N must be a scalar integer> hilb (ones (2)) |