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1 ## Copyright (C) 2000, 2006, 2007 Paul Kienzle |
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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 3 of the License, or (at |
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8 ## your option) 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, see |
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17 ## <http://www.gnu.org/licenses/>. |
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18 |
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19 ## -*- texinfo -*- |
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20 ## @deftypefn {Function File} {[@var{r}, @var{k}] =} rref (@var{a}, @var{tol}) |
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21 ## |
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22 ## Returns the reduced row echelon form of @var{a}. @var{tol} defaults |
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23 ## to @code{eps * max (size (@var{a})) * norm (@var{a}, inf)}. |
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24 ## |
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25 ## Called with two return arguments, @var{k} returns the vector of |
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26 ## "bound variables", which are those columns on which elimination |
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27 ## has been performed. |
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28 ## |
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29 ## @end deftypefn |
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30 |
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31 ## Author: Paul Kienzle <pkienzle@users.sf.net> |
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32 ## (based on a anonymous source from the public domain) |
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33 |
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34 function [A, k] = rref (A, tolerance) |
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35 |
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36 if (nargin < 1 || nargin > 2) |
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37 print_usage (); |
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38 endif |
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39 |
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40 if (ndims (A) > 2) |
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41 error ("rref: expecting matrix argument"); |
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42 endif |
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43 |
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44 [rows, cols] = size (A); |
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45 |
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46 if (nargin < 2) |
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47 tolerance = eps * max (rows, cols) * norm (A, inf); |
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48 endif |
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49 |
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50 used = zeros (1, cols); |
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51 r = 1; |
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52 for c = 1:cols |
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53 ## Find the pivot row |
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54 [m, pivot] = max (abs (A(r:rows,c))); |
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55 pivot = r + pivot - 1; |
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56 |
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57 if (m <= tolerance) |
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58 ## Skip column c, making sure the approximately zero terms are |
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59 ## actually zero. |
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60 A (r:rows, c) = zeros (rows-r+1, 1); |
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61 else |
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62 ## keep track of bound variables |
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63 used (1, c) = 1; |
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64 |
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65 ## Swap current row and pivot row |
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66 A ([pivot, r], c:cols) = A ([r, pivot], c:cols); |
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67 |
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68 ## Normalize pivot row |
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69 A (r, c:cols) = A (r, c:cols) / A (r, c); |
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70 |
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71 ## Eliminate the current column |
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72 ridx = [1:r-1, r+1:rows]; |
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73 A (ridx, c:cols) = A (ridx, c:cols) - A (ridx, c) * A(r, c:cols); |
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74 |
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75 ## Check if done |
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76 if (r++ == rows) |
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77 break; |
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78 endif |
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79 endif |
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80 endfor |
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81 k = find (used); |
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82 |
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83 endfunction |
