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hilb.m: Speed-up function using broadcasting. * scripts/special-matrix/hilb.m: Speed-up function using broadcasting. For more information read https://octave.discourse.group/t/boosting-hilb/48 .
author Kai T. Ohlhus <k.ohlhus@gmail.com>
date Wed, 01 Jul 2020 16:03:41 +0900
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1 ########################################################################
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2 ##
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3 ## Copyright (C) 1993-2020 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/>.
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7 ##
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8 ## This file is part of Octave.
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9 ##
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10 ## Octave is free software: you can redistribute it and/or modify it
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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.
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14 ##
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15 ## Octave is distributed in the hope that it will be useful, but
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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.
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19 ##
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20 ## You should have received a copy of the GNU General Public License
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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 ########################################################################
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26 ## -*- texinfo -*-
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27 ## @deftypefn {} {} 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
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31 ## @tex
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32 ## $$
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33 ## H(i, j) = {1 \over (i + j - 1)}
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34 ## $$
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35 ## @end tex
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36 ## @ifnottex
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37 ##
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38 ## @example
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39 ## H(i, j) = 1 / (i + j - 1)
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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}
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59 ## @end deftypefn
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60
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61 function retval = hilb (n)
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62
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63 if (nargin != 1)
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64 print_usage ();
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65 elseif (! isscalar (n))
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66 error ("hilb: N must be a scalar integer");
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67 endif
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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 retval = 1 ./ (j' + j - 1);
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73
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74 endfunction
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75
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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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79
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80 %!error hilb ()
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81 %!error hilb (1, 2)
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82 %!error <N must be a scalar integer> hilb (ones (2))