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## Copyright (C) 2000-2013 Paul Kienzle
##
## This file is part of Octave.
##
## Octave is free software; you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 3 of the License, or (at
## your option) any later version.
##
## Octave is distributed in the hope that it will be useful, but
## WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
## General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, see
## <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {} cplxpair (@var{z})
## @deftypefnx {Function File} {} cplxpair (@var{z}, @var{tol})
## @deftypefnx {Function File} {} cplxpair (@var{z}, @var{tol}, @var{dim})
## Sort the numbers @var{z} into complex conjugate pairs ordered by
## increasing real part. Place the negative imaginary complex number
## first within each pair. Place all the real numbers (those with
## @code{abs (imag (@var{z}) / @var{z}) < @var{tol})}) after the
## complex pairs.
##
## If @var{tol} is unspecified the default value is 100*@code{eps}.
##
## By default the complex pairs are sorted along the first non-singleton
## dimension of @var{z}. If @var{dim} is specified, then the complex
## pairs are sorted along this dimension.
##
## Signal an error if some complex numbers could not be paired. Signal an
## error if all complex numbers are not exact conjugates (to within
## @var{tol}). Note that there is no defined order for pairs with identical
## real parts but differing imaginary parts.
## @c Set example in small font to prevent overfull line
##
## @smallexample
## cplxpair (exp(2i*pi*[0:4]'/5)) == exp(2i*pi*[3; 2; 4; 1; 0]/5)
## @end smallexample
## @end deftypefn
## FIXME: subsort returned pairs by imaginary magnitude
## FIXME: Why doesn't exp (2i*pi*[0:4]'/5) produce exact conjugates. Does
## FIXME: it in Matlab? The reason is that complex pairs are supposed
## FIXME: to be exact conjugates, and not rely on a tolerance test.
## 2006-05-12 David Bateman - Modified for NDArrays
function y = cplxpair (z, tol, dim)
if (nargin < 1 || nargin > 3)
print_usage ();
endif
if (length (z) == 0)
y = zeros (size (z));
return;
endif
if (nargin < 2 || isempty (tol))
if (isa (z, "single"))
tol = 100 * eps("single");
else
tol = 100*eps;
endif
endif
nd = ndims (z);
orig_dims = size (z);
if (nargin < 3)
## Find the first singleton dimension.
dim = 0;
while (dim < nd && orig_dims(dim+1) == 1)
dim++;
endwhile
dim++;
if (dim > nd)
dim = 1;
endif
else
dim = floor (dim);
if (dim < 1 || dim > nd)
error ("cplxpair: invalid dimension along which to sort");
endif
endif
## Move dimension to treat first, and convert to a 2-D matrix.
perm = [dim:nd, 1:dim-1];
z = permute (z, perm);
sz = size (z);
n = sz (1);
m = prod (sz) / n;
z = reshape (z, n, m);
## Sort the sequence in terms of increasing real values.
[q, idx] = sort (real (z), 1);
z = z(idx + n * ones (n, 1) * [0:m-1]);
## Put the purely real values at the end of the returned list.
cls = "double";
if (isa (z, "single"))
cls = "single";
endif
[idxi, idxj] = find (abs (imag (z)) ./ (abs (z) + realmin (cls)) < tol);
q = sparse (idxi, idxj, 1, n, m);
nr = sum (q, 1);
[q, idx] = sort (q, 1);
z = z(idx);
y = z;
## For each remaining z, place the value and its conjugate at the
## start of the returned list, and remove them from further
## consideration.
for j = 1:m
p = n - nr(j);
for i = 1:2:p
if (i+1 > p)
error ("cplxpair: could not pair all complex numbers");
endif
[v, idx] = min (abs (z(i+1:p) - conj (z(i))));
if (v > tol)
error ("cplxpair: could not pair all complex numbers");
endif
if (imag (z(i)) < 0)
y([i, i+1]) = z([i, idx+i]);
else
y([i, i+1]) = z([idx+i, i]);
endif
z(idx+i) = z(i+1);
endfor
endfor
## Reshape the output matrix.
y = ipermute (reshape (y, sz), perm);
endfunction
%!demo
%! [ cplxpair(exp(2i*pi*[0:4]'/5)), exp(2i*pi*[3; 2; 4; 1; 0]/5) ]
%!assert (isempty (cplxpair ([])))
%!assert (cplxpair (1), 1)
%!assert (cplxpair ([1+1i, 1-1i]), [1-1i, 1+1i])
%!assert (cplxpair ([1+1i, 1+1i, 1, 1-1i, 1-1i, 2]), ...
%! [1-1i, 1+1i, 1-1i, 1+1i, 1, 2])
%!assert (cplxpair ([1+1i; 1+1i; 1; 1-1i; 1-1i; 2]), ...
%! [1-1i; 1+1i; 1-1i; 1+1i; 1; 2])
%!assert (cplxpair ([0, 1, 2]), [0, 1, 2])
%!shared z
%! z = exp (2i*pi*[4; 3; 5; 2; 6; 1; 0]/7);
%!assert (cplxpair (z(randperm (7))), z)
%!assert (cplxpair (z(randperm (7))), z)
%!assert (cplxpair (z(randperm (7))), z)
%!assert (cplxpair ([z(randperm(7)),z(randperm(7))]), [z,z])
%!assert (cplxpair ([z(randperm(7)),z(randperm(7))],[],1), [z,z])
%!assert (cplxpair ([z(randperm(7)).';z(randperm(7)).'],[],2), [z.';z.'])
## tolerance test
%!assert (cplxpair ([1i, -1i, 1+(1i*eps)],2*eps), [-1i, 1i, 1+(1i*eps)])
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