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1 ## Copyright (C) 1996, 1997 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 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} {} null (@var{a}, @var{tol}) |
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22 ## Return an orthonormal basis of the null space of @var{a}. |
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23 ## |
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24 ## The dimension of the null space is taken as the number of singular |
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25 ## values of @var{a} not greater than @var{tol}. If the argument @var{tol} |
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26 ## is missing, it is computed as |
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27 ## |
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28 ## @example |
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29 ## max (size (@var{a})) * max (svd (@var{a})) * eps |
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30 ## @end example |
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31 ## @end deftypefn |
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32 |
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33 ## Author: KH <Kurt.Hornik@wu-wien.ac.at> |
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34 ## Created: 24 December 1993. |
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35 ## Adapted-By: jwe |
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36 |
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37 function retval = null (A, tol) |
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38 |
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39 if (isempty (A)) |
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40 retval = []; |
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41 else |
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42 [U, S, V] = svd (A); |
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43 |
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44 [rows, cols] = size (A); |
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45 |
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46 [S_nr, S_nc] = size (S); |
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47 |
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48 if (S_nr == 1 || S_nc == 1) |
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49 s = S(1); |
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50 else |
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51 s = diag (S); |
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52 endif |
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53 |
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54 if (nargin == 1) |
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55 tol = max (size (A)) * s (1) * eps; |
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56 elseif (nargin != 2) |
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57 usage ("null (A, tol)"); |
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58 endif |
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59 |
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60 rank = sum (s > tol); |
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61 |
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62 if (rank < cols) |
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63 retval = V (:, rank+1:cols); |
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64 else |
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65 retval = zeros (cols, 0); |
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66 endif |
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67 endif |
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68 |
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69 endfunction |
