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1 # Copyright (C) 1995 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 the |
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7 # Free Software Foundation; either version 2, or (at your option) any |
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8 # later version. |
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9 # |
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10 # Octave is distributed in the hope that it will be useful, but WITHOUT |
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11 # ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
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12 # FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
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13 # 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, 675 Mass Ave, Cambridge, MA 02139, USA. |
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18 |
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19 function retval = null (A, tol) |
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20 |
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21 # usage: null (A, tol) |
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22 # null (A) |
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23 # |
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24 # Returns an orthonormal basis of the null space of A. |
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25 # |
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26 # The dimension of the null space is taken as the number of singular |
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27 # values of A not greater than tol; the default for tol is |
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28 # max (size (A)) * sigma_max (A) * eps, where sigma_max (A) is the |
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29 # maximal singular value of A. |
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30 |
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31 # Written by KH (Kurt.Hornik@ci.tuwien.ac.at) on Dec 24, 1993. |
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32 |
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33 [U, S, V] = svd (A); |
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34 |
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35 [rows, cols] = size (A); |
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36 |
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37 s = diag (S); |
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38 |
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39 if (nargin == 1) |
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40 tol = max (size (A)) * s (1) * eps; |
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41 else (nargin != 2) |
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42 usage ("null (A [, tol])"); |
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43 endif |
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44 |
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45 rank = sum (s > tol); |
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46 |
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47 if (rank < cols) |
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48 retval = V (:, rank+1:cols); |
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49 else |
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50 retval = zeros (cols, 0); |
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51 endif |
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52 |
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53 endfunction |
