scripts/general/cplxpair.m

## Copyright (C) 2000, 2006, 2007, 2008, 2009 Paul Kienzle
##
## This file is part of Octave.
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## under the terms of the GNU General Public License as published by
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## General Public License for more details.
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## along with Octave; see the file COPYING.  If not, see
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## -*- 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)]);