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[project @ 2006-05-17 21:00:54 by jwe]
author jwe
date Wed, 17 May 2006 21:00:54 +0000
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5819:e54c11df0524 5820:27c966e4b2dc
1 ## Copyright (C) 2000 Paul Kienzle
2 ##
3 ## This file is part of Octave.
4 ##
5 ## Octave is free software; you can redistribute it and/or modify it
6 ## under the terms of the GNU General Public License as published by
7 ## the Free Software Foundation; either version 2, or (at your option)
8 ## any later version.
9 ##
10 ## Octave is distributed in the hope that it will be useful, but
11 ## WITHOUT ANY WARRANTY; without even the implied warranty of
12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 ## General Public License for more details.
14 ##
15 ## You should have received a copy of the GNU General Public License
16 ## along with Octave; see the file COPYING. If not, write to the Free
17 ## Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
18 ## 02110-1301, USA.
19
20 ## -*- texinfo -*-
21 ## @deftypefn {Function File} {} cplxpair (@var{z}, @var{tol}, @var{dim})
22 ## Sort the numbers @var{z} into complex conjugate pairs ordered by
23 ## increasing real part. With identical real parts, order by increasing
24 ## imaginary magnitude. Place the negative imaginary complex number
25 ## first within each pair. Place all the real numbers after all the
26 ## complex pairs (those with @code {abs ( imag (@var{z}) / @var{z}) <
27 ## @var{tol}}), where the default value of @var{tol} is @code{100 *
28 ## @var{eps}}.
29 ##
30 ## By default the complex pairs are sorted along the first non-singleton
31 ## dimension of @var{z}. If @var{dim} is specified, then the complex
32 ## pairs are sorted along this dimension.
33 ##
34 ## Signal an error if some complex numbers could not be paired. Requires
35 ## all complex numbers to be exact conjugates within tol, or signals an
36 ## error. Note that there are no guarantees on the order of the returned
37 ## pairs with identical real parts but differing imaginary parts.
38 ##
39 ## @example
40 ## cplxpair (exp(2i*pi*[0:4]'/5)) == exp(2i*pi*[3; 2; 4; 1; 0]/5)
41 ## @end example
42 ## @end deftypefn
43
44 ## TODO: subsort returned pairs by imaginary magnitude
45 ## TODO: Why doesn't exp(2i*pi*[0:4]'/5) produce exact conjugates. Does
46 ## TODO: it in Matlab? The reason is that complex pairs are supposed
47 ## TODO: to be exact conjugates, and not rely on a tolerance test.
48
49 ## 2006-05-12 David Bateman - Modified for NDArrays
50
51 function y = cplxpair (z, tol, dim)
52
53 if nargin < 1 || nargin > 3
54 usage ("z = cplxpair (z, tol, dim);");
55 endif
56
57 if (length (z) == 0)
58 y = zeros (size (z));
59 return;
60 endif
61
62 if (nargin < 2 || isempty (tol))
63 tol = 100*eps;
64 endif
65
66 nd = ndims (z);
67 orig_dims = size (z);
68 if (nargin < 3)
69 ## Find the first singleton dimension.
70 dim = 0;
71 while (dim < nd && orig_dims(dim+1) == 1)
72 dim++;
73 endwhile
74 dim++;
75 if (dim > nd)
76 dim = 1;
77 endif
78 else
79 dim = floor(dim);
80 if (dim < 1 || dim > nd)
81 error ("cplxpair: invalid dimension along which to sort");
82 endif
83 endif
84
85 ## Move dimension to treat first, and convert to a 2-D matrix
86 perm = [dim:nd, 1:dim-1];
87 z = permute (z, perm);
88 sz = size (z);
89 n = sz (1);
90 m = prod (sz) / n;
91 z = reshape (z, n, m);
92
93 ## Sort the sequence in terms of increasing real values
94 [q, idx] = sort (real (z), 1);
95 z = z(idx + n * ones (n, 1) * [0:m-1]);
96
97 ## Put the purely real values at the end of the returned list
98 [idxi, idxj] = find (abs (imag (z)) ./ (abs (z) + realmin) < tol );
99 q = sparse (idxi, idxj, 1, n, m);
100 nr = sum (q, 1);
101 [q, idx] = sort (q, 1);
102 z = z(idx);
103 y = z;
104
105 ## For each remaining z, place the value and its conjugate at the
106 ## start of the returned list, and remove them from further
107 ## consideration.
108 for j = 1:m
109 p = n - nr(j);
110 for i=1:2:p
111 if (i+1 > p)
112 error ("cplxpair could not pair all complex numbers");
113 endif
114 [v, idx] = min (abs (z(i+1:p) - conj (z(i))));
115 if (v > tol)
116 error ("cplxpair could not pair all complex numbers");
117 endif
118 if (imag (z(i)) < 0)
119 y([i, i+1]) = z([i, idx+i]);
120 else
121 y([i, i+1]) = z([idx+i, i]);
122 endif
123 z(idx+i) = z(i+1);
124 endfor
125 endfor
126
127 ## Reshape the output matrix
128 y = ipermute (reshape (y, sz), perm);
129
130 endfunction
131
132 %!demo
133 %! [ cplxpair(exp(2i*pi*[0:4]'/5)), exp(2i*pi*[3; 2; 4; 1; 0]/5) ]
134
135 %!assert (isempty(cplxpair([])));
136 %!assert (cplxpair(1), 1)
137 %!assert (cplxpair([1+1i, 1-1i]), [1-1i, 1+1i])
138 %!assert (cplxpair([1+1i, 1+1i, 1, 1-1i, 1-1i, 2]), \
139 %! [1-1i, 1+1i, 1-1i, 1+1i, 1, 2])
140 %!assert (cplxpair([1+1i; 1+1i; 1; 1-1i; 1-1i; 2]), \
141 %! [1-1i; 1+1i; 1-1i; 1+1i; 1; 2])
142 %!assert (cplxpair([0, 1, 2]), [0, 1, 2]);
143
144 %!shared z
145 %! z=exp(2i*pi*[4; 3; 5; 2; 6; 1; 0]/7);
146 %!assert (cplxpair(z(randperm(7))), z);
147 %!assert (cplxpair(z(randperm(7))), z);
148 %!assert (cplxpair(z(randperm(7))), z);
149 %!assert (cplxpair([z(randperm(7)),z(randperm(7))]),[z,z])
150 %!assert (cplxpair([z(randperm(7)),z(randperm(7))],[],1),[z,z])
151 %!assert (cplxpair([z(randperm(7)).';z(randperm(7)).'],[],2),[z.';z.'])
152
153 %!## tolerance test
154 %!assert (cplxpair([1i, -1i, 1+(1i*eps)],2*eps), [-1i, 1i, 1+(1i*eps)]);