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doc: Review and update documentation for "Matrix Manipulation" chapter. * matrix.txi, arrayfun.m, blkdiag.m, fliplr.m, flipud.m, logspace.m, postpad.m, prepad.m, randi.m, repmat.m, rot90.m, rotdim.m, shiftdim.m, sortrows.m, vech.m, xor.m, hadamard.m, hankel.m, hilb.m, invhilb.m, magic.m, pascal.m, rosser.m, sylvester_matrix.m, toeplitz.m, vander.m, wilkinson.m, bsxfun.cc, find.cc, lookup.cc, rand.cc, tril.cc, data.cc, arrayfun.m, blkdiag.m, fliplr.m, flipud.m, logspace.m, postpad.m, prepad.m, randi.m, repmat.m, rot90.m, rotdim.m, shiftdim.m, sortrows.m, vech.m, xor.m, hadamard.m, hankel.m, hilb.m, invhilb.m, magic.m, pascal.m, rosser.m, sylvester_matrix.m, toeplitz.m, vander.m, wilkinson.m, bsxfun.cc (bsxfun), find.cc (find), lookup.cc (lookup), rand.cc (rand, randn, rande, randg, randp), tril.cc (triu), data.cc (all, any, horzcat, vertcat, cat, permute, ipermute, ones, zeros, eye, linspace, resize, reshape, issorted, diff): Improve docstrings
author Rik <octave@nomad.inbox5.com>
date Sun, 01 May 2011 08:55:15 -0700
parents fd0a3ac60b0e
children 779e15b69738
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John W. Eaton <jwe@octave.org>
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1 ## Copyright (C) 1993-2011 John W. Eaton
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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
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19 ## -*- texinfo -*-
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20 ## @deftypefn {Function File} {} hilb (@var{n})
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21 ## Return the Hilbert matrix of order @var{n}. The @math{i,j} element
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22 ## of a Hilbert matrix is defined as
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23 ## @tex
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24 ## $$
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25 ## H (i, j) = {1 \over (i + j - 1)}
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26 ## $$
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27 ## @end tex
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28 ## @ifnottex
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29 ##
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30 ## @example
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31 ## H (i, j) = 1 / (i + j - 1)
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32 ## @end example
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33 ##
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34 ## @end ifnottex
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35 ##
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36 ## Hilbert matrices are close to being singular which make them difficult to
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37 ## invert with numerical routines.
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38 ## Comparing the condition number of a random matrix 5x5 matrix with that of
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39 ## a Hilbert matrix of order 5 reveals just how difficult the problem is.
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40 ##
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41 ## @example
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42 ## @group
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43 ## cond (rand (5))
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44 ## @result{} 14.392
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45 ## cond (hilb (5))
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46 ## @result{} 4.7661e+05
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47 ## @end group
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48 ## @end example
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49 ##
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50 ## @seealso{invhilb}
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51 ## @end deftypefn
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53 ## Author: jwe
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54
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55 function retval = hilb (n)
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57
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58 if (nargin != 1)
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59 print_usage ();
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60 endif
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61
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62 nmax = length (n);
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63 if (nmax == 1)
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64 retval = zeros (n);
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65 tmp = 1:n;
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66 for i = 1:n
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67 retval (i, :) = 1.0 ./ (tmp + (i - 1));
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68 endfor
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69 else
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70 error ("hilb: expecting scalar argument, found something else");
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71 endif
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72
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73 endfunction
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75 %!assert((hilb (2) == [1, 1/2; 1/2, 1/3]
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76 %! && hilb (3) == [1, 1/2, 1/3; 1/2, 1/3, 1/4; 1/3, 1/4, 1/5]));
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78 %!error hilb ();
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80 %!error hilb (1, 2);
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81