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1 ## Copyright (C) 1995, 1996 Kurt Hornik |
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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 2, or (at your option) |
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8 ## 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, write to the Free |
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17 ## Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA |
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18 ## 02110-1301, USA. |
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19 |
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20 ## -*- texinfo -*- |
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21 ## @deftypefn {Function File} {} duplication_matrix (@var{n}) |
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22 ## Return the duplication matrix |
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23 ## @iftex |
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24 ## @tex |
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25 ## $D_n$ |
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26 ## @end tex |
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27 ## @end iftex |
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28 ## @ifinfo |
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29 ## @math{Dn} |
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30 ## @end ifinfo |
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31 ## which is the unique |
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32 ## @iftex |
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33 ## @tex |
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34 ## $n^2 \times n(n+1)/2$ |
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35 ## @end tex |
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36 ## @end iftex |
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37 ## @ifinfo |
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38 ## @math{n^2} by @math{n*(n+1)/2} |
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39 ## @end ifinfo |
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40 ## matrix such that |
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41 ## @iftex |
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42 ## @tex |
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43 ## $D_n * {\rm vech} (A) = {\rm vec} (A)$ |
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44 ## @end tex |
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45 ## @end iftex |
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46 ## @ifinfo |
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47 ## @math{Dn vech (A) = vec (A)} |
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48 ## @end ifinfo |
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49 ## for all symmetric |
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50 ## @iftex |
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51 ## @tex |
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52 ## $n \times n$ |
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53 ## @end tex |
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54 ## @end iftex |
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55 ## @ifinfo |
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56 ## @math{n} by @math{n} |
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57 ## @end ifinfo |
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58 ## matrices |
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59 ## @iftex |
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60 ## @tex |
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61 ## $A$. |
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62 ## @end tex |
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63 ## @end iftex |
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64 ## @ifinfo |
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65 ## @math{A}. |
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66 ## @end ifinfo |
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67 ## |
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68 ## See Magnus and Neudecker (1988), Matrix differential calculus with |
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69 ## applications in statistics and econometrics. |
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70 ## @end deftypefn |
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71 |
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72 ## Author: KH <Kurt.Hornik@wu-wien.ac.at> |
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73 ## Created: 8 May 1995 |
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74 ## Adapged-By: jwe |
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75 |
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76 function d = duplication_matrix (n) |
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77 |
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78 if (nargin != 1) |
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79 print_usage (); |
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80 endif |
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81 |
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82 if (! (isscalar (n) && n == round (n) && n > 0)) |
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83 error ("duplication_matrix: n must be a positive integer"); |
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84 endif |
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85 |
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86 d = zeros (n * n, n * (n + 1) / 2); |
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87 |
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88 ## It is clearly possible to make this a LOT faster! |
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89 count = 0; |
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90 for j = 1 : n |
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91 d ((j - 1) * n + j, count + j) = 1; |
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92 for i = (j + 1) : n |
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93 d ((j - 1) * n + i, count + i) = 1; |
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94 d ((i - 1) * n + j, count + i) = 1; |
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95 endfor |
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96 count = count + n - j; |
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97 endfor |
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98 |
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99 endfunction |
