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1 ## Copyright (C) 2000 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 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} {} cplxpair (@var{z}, @var{tol}, @var{dim}) |
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22 ## Sort the numbers @var{z} into complex conjugate pairs ordered by |
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23 ## increasing real part. With identical real parts, order by increasing |
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24 ## imaginary magnitude. Place the negative imaginary complex number |
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25 ## first within each pair. Place all the real numbers after all the |
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26 ## complex pairs (those with @code {abs ( imag (@var{z}) / @var{z}) < |
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27 ## @var{tol}}), where the default value of @var{tol} is @code{100 * |
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28 ## @var{eps}}. |
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29 ## |
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30 ## By default the complex pairs are sorted along the first non-singleton |
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31 ## dimension of @var{z}. If @var{dim} is specified, then the complex |
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32 ## pairs are sorted along this dimension. |
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33 ## |
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34 ## Signal an error if some complex numbers could not be paired. Requires |
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35 ## all complex numbers to be exact conjugates within tol, or signals an |
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36 ## error. Note that there are no guarantees on the order of the returned |
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37 ## pairs with identical real parts but differing imaginary parts. |
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38 ## |
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39 ## @example |
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40 ## cplxpair (exp(2i*pi*[0:4]'/5)) == exp(2i*pi*[3; 2; 4; 1; 0]/5) |
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41 ## @end example |
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42 ## @end deftypefn |
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43 |
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44 ## TODO: subsort returned pairs by imaginary magnitude |
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45 ## TODO: Why doesn't exp(2i*pi*[0:4]'/5) produce exact conjugates. Does |
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46 ## TODO: it in Matlab? The reason is that complex pairs are supposed |
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47 ## TODO: to be exact conjugates, and not rely on a tolerance test. |
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48 |
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49 ## 2006-05-12 David Bateman - Modified for NDArrays |
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50 |
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51 function y = cplxpair (z, tol, dim) |
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52 |
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53 if nargin < 1 || nargin > 3 |
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54 usage ("z = cplxpair (z, tol, dim);"); |
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55 endif |
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56 |
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57 if (length (z) == 0) |
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58 y = zeros (size (z)); |
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59 return; |
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60 endif |
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61 |
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62 if (nargin < 2 || isempty (tol)) |
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63 tol = 100*eps; |
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64 endif |
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65 |
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66 nd = ndims (z); |
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67 orig_dims = size (z); |
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68 if (nargin < 3) |
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69 ## Find the first singleton dimension. |
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70 dim = 0; |
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71 while (dim < nd && orig_dims(dim+1) == 1) |
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72 dim++; |
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73 endwhile |
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74 dim++; |
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75 if (dim > nd) |
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76 dim = 1; |
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77 endif |
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78 else |
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79 dim = floor(dim); |
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80 if (dim < 1 || dim > nd) |
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81 error ("cplxpair: invalid dimension along which to sort"); |
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82 endif |
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83 endif |
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84 |
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85 ## Move dimension to treat first, and convert to a 2-D matrix |
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86 perm = [dim:nd, 1:dim-1]; |
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87 z = permute (z, perm); |
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88 sz = size (z); |
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89 n = sz (1); |
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90 m = prod (sz) / n; |
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91 z = reshape (z, n, m); |
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92 |
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93 ## Sort the sequence in terms of increasing real values |
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94 [q, idx] = sort (real (z), 1); |
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95 z = z(idx + n * ones (n, 1) * [0:m-1]); |
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96 |
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97 ## Put the purely real values at the end of the returned list |
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98 [idxi, idxj] = find (abs (imag (z)) ./ (abs (z) + realmin) < tol ); |
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99 q = sparse (idxi, idxj, 1, n, m); |
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100 nr = sum (q, 1); |
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101 [q, idx] = sort (q, 1); |
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102 z = z(idx); |
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103 y = z; |
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104 |
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105 ## For each remaining z, place the value and its conjugate at the |
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106 ## start of the returned list, and remove them from further |
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107 ## consideration. |
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108 for j = 1:m |
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109 p = n - nr(j); |
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110 for i=1:2:p |
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111 if (i+1 > p) |
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112 error ("cplxpair could not pair all complex numbers"); |
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113 endif |
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114 [v, idx] = min (abs (z(i+1:p) - conj (z(i)))); |
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115 if (v > tol) |
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116 error ("cplxpair could not pair all complex numbers"); |
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117 endif |
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118 if (imag (z(i)) < 0) |
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119 y([i, i+1]) = z([i, idx+i]); |
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120 else |
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121 y([i, i+1]) = z([idx+i, i]); |
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122 endif |
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123 z(idx+i) = z(i+1); |
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124 endfor |
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125 endfor |
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126 |
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127 ## Reshape the output matrix |
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128 y = ipermute (reshape (y, sz), perm); |
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129 |
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130 endfunction |
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131 |
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132 %!demo |
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133 %! [ cplxpair(exp(2i*pi*[0:4]'/5)), exp(2i*pi*[3; 2; 4; 1; 0]/5) ] |
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134 |
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135 %!assert (isempty(cplxpair([]))); |
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136 %!assert (cplxpair(1), 1) |
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137 %!assert (cplxpair([1+1i, 1-1i]), [1-1i, 1+1i]) |
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138 %!assert (cplxpair([1+1i, 1+1i, 1, 1-1i, 1-1i, 2]), \ |
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139 %! [1-1i, 1+1i, 1-1i, 1+1i, 1, 2]) |
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140 %!assert (cplxpair([1+1i; 1+1i; 1; 1-1i; 1-1i; 2]), \ |
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141 %! [1-1i; 1+1i; 1-1i; 1+1i; 1; 2]) |
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142 %!assert (cplxpair([0, 1, 2]), [0, 1, 2]); |
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143 |
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144 %!shared z |
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145 %! z=exp(2i*pi*[4; 3; 5; 2; 6; 1; 0]/7); |
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146 %!assert (cplxpair(z(randperm(7))), z); |
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147 %!assert (cplxpair(z(randperm(7))), z); |
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148 %!assert (cplxpair(z(randperm(7))), z); |
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149 %!assert (cplxpair([z(randperm(7)),z(randperm(7))]),[z,z]) |
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150 %!assert (cplxpair([z(randperm(7)),z(randperm(7))],[],1),[z,z]) |
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151 %!assert (cplxpair([z(randperm(7)).';z(randperm(7)).'],[],2),[z.';z.']) |
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152 |
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153 %!## tolerance test |
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154 %!assert (cplxpair([1i, -1i, 1+(1i*eps)],2*eps), [-1i, 1i, 1+(1i*eps)]); |
