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1 ## Copyright (C) 1996 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, 59 Temple Place - Suite 330, Boston, MA |
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18 ## 02111-1307, USA. |
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19 |
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20 ## usage: null (A, tol) |
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21 ## null (A) |
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22 ## |
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23 ## Returns an orthonormal basis of the null space of A. |
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24 ## |
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25 ## The dimension of the null space is taken as the number of singular |
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26 ## values of A not greater than tol; the default for tol is |
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27 ## max (size (A)) * sigma_max (A) * eps, where sigma_max (A) is the |
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28 ## maximal singular value of A. |
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29 |
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30 ## Author: KH <Kurt.Hornik@ci.tuwien.ac.at> |
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31 ## Created: 24 December 1993. |
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32 ## Adapted-By: jwe |
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33 |
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34 function retval = null (A, tol) |
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35 |
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36 [U, S, V] = svd (A); |
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37 |
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38 [rows, cols] = size (A); |
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39 |
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40 [S_nr, S_nc] = size (S); |
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41 |
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42 if (S_nr == 1 || S_nc == 1) |
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43 s = S(1); |
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44 else |
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45 s = diag (S); |
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46 endif |
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47 |
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48 if (nargin == 1) |
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49 tol = max (size (A)) * s (1) * eps; |
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50 elseif (nargin != 2) |
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51 usage ("null (A [, tol])"); |
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52 endif |
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53 |
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54 rank = sum (s > tol); |
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55 |
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56 if (rank < cols) |
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57 retval = V (:, rank+1:cols); |
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58 else |
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59 retval = zeros (cols, 0); |
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60 endif |
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61 |
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62 endfunction |
