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maint: update copyright notices for 2012
author John W. Eaton <jwe@octave.org>
date Mon, 02 Jan 2012 14:25:41 -0500
parents c5c94b63931f
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14138
72c96de7a403 maint: update copyright notices for 2012
John W. Eaton <jwe@octave.org>
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1 ## Copyright (C) 2000-2012 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} {} rref (@var{A})
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21 ## @deftypefnx {Function File} {} rref (@var{A}, @var{tol})
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22 ## @deftypefnx {Function File} {[@var{r}, @var{k}] =} rref (@dots{})
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23 ## Return the reduced row echelon form of @var{A}. @var{tol} defaults
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24 ## to @code{eps * max (size (@var{A})) * norm (@var{A}, inf)}.
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25 ##
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26 ## Called with two return arguments, @var{k} returns the vector of
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27 ## "bound variables", which are those columns on which elimination
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28 ## has been performed.
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29 ##
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30 ## @end deftypefn
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31
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32 ## Author: Paul Kienzle <pkienzle@users.sf.net>
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33 ## (based on an anonymous source from the public domain)
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34
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35 function [A, k] = rref (A, tol)
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36
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37 if (nargin < 1 || nargin > 2)
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38 print_usage ();
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39 endif
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40
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41 if (ndims (A) > 2)
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42 error ("rref: expecting matrix argument");
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43 endif
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44
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45 [rows, cols] = size (A);
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46
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47 if (nargin < 2)
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48 if (isa (A, "single"))
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49 tol = eps ("single") * max (rows, cols) * norm (A, inf ("single"));
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50 else
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51 tol = eps * max (rows, cols) * norm (A, inf);
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52 endif
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53 endif
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54
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55 used = zeros (1, cols);
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56 r = 1;
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57 for c = 1:cols
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58 ## Find the pivot row
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59 [m, pivot] = max (abs (A(r:rows,c)));
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60 pivot = r + pivot - 1;
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61
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62 if (m <= tol)
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63 ## Skip column c, making sure the approximately zero terms are
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64 ## actually zero.
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65 A (r:rows, c) = zeros (rows-r+1, 1);
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66 else
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67 ## keep track of bound variables
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68 used (1, c) = 1;
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69
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70 ## Swap current row and pivot row
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71 A ([pivot, r], c:cols) = A ([r, pivot], c:cols);
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72
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73 ## Normalize pivot row
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74 A (r, c:cols) = A (r, c:cols) / A (r, c);
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75
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76 ## Eliminate the current column
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77 ridx = [1:r-1, r+1:rows];
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78 A (ridx, c:cols) = A (ridx, c:cols) - A (ridx, c) * A(r, c:cols);
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79
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80 ## Check if done
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81 if (r++ == rows)
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82 break;
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83 endif
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84 endif
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85 endfor
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86 k = find (used);
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87
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88 endfunction
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90 %!test
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91 %! a = [1];
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92 %! [r k] = rref(a);
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93 %! assert(r, [1], 2e-8);
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94 %! assert(k, [1], 2e-8);
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95
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96 %!test
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97 %! a = [1 3; 4 5];
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98 %! [r k] = rref(a);
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99 %! assert(rank(a), rank(r), 2e-8);
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100 %! assert(r, eye(2), 2e-8);
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101 %! assert(k == [1, 2] || k == [2, 1]);
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103
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104 %!test
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105 %! a = [1 3; 4 5; 7 9];
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106 %! [r k] = rref(a);
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107 %! assert(rank(a), rank(r), 2e-8);
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108 %! assert(r, eye(3)(:,1:2), 2e-8);
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109 %! assert(k, [1 2], 2e-8);
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110
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111 %!test
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112 %! a = [1 2 3; 2 4 6; 7 2 0];
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113 %! [r k] = rref(a);
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114 %! assert(rank(a), rank(r), 2e-8);
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115 %! assert(r, [1 0 (3-7/2); 0 1 (7/4); 0 0 0], 2e-8);
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116 %! assert(k, [1 2], 2e-8);
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117
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118 %!test
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119 %! a = [1 2 1; 2 4 2.01; 2 4 2.1];
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120 %! tol = 0.02;
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121 %! [r k] = rref(a, tol);
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122 %! assert(rank(a, tol), rank(r, tol), 2e-8);
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123 %! tol = 0.2;
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124 %! [r k] = rref(a, tol);
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125 %! assert(rank(a, tol), rank(r, tol), 2e-8);
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126
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127 %!error rref();
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