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+## Copyright (C) 2004-2012 David Bateman and Andy Adler
+## Copyright (C) 2012 Jordi Gutiérrez Hermoso
+##
+## 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
+## .
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {} sprandsym (@var{n}, @var{d})
+## @deftypefnx {Function File} {} sprandsym (@var{s})
+## Generate a symmetric random sparse matrix. The size of the matrix will be
+## @var{n} by @var{n}, with a density of values given by @var{d}.
+## @var{d} should be between 0 and 1. Values will be normally
+## distributed with mean of zero and variance 1.
+##
+## If called with a single matrix argument, a random sparse matrix is
+## generated wherever the matrix @var{S} is non-zero in its lower
+## triangular part.
+## @seealso{sprand, sprandn}
+## @end deftypefn
+
+function S = sprandsym (n, d)
+
+ if (nargin != 1 && nargin != 2)
+ print_usage ();
+ endif
+
+ if (nargin == 1)
+ [i, j] = find (tril (n));
+ [nr, nc] = size (n);
+ S = sparse (i, j, randn (size (i)), nr, nc);
+ S = S + tril (S, -1)';
+ return;
+ endif
+
+ if (!(isscalar (n) && n == fix (n) && n > 0))
+ error ("sprandsym: N must be an integer greater than 0");
+ endif
+
+ if (d < 0 || d > 1)
+ error ("sprandsym: density D must be between 0 and 1");
+ endif
+
+ ## Actual number of nonzero entries
+ k = round (n^2*d);
+
+ ## Diagonal nonzero entries, same parity as k
+ r = pick_rand_diag (n, k);
+
+ ## Off diagonal nonzero entries
+ m = (k - r)/2;
+
+ ondiag = randperm (n, r);
+ offdiag = randperm (n*(n - 1)/2, m);
+
+ ## Row index
+ i = lookup (cumsum (0:n), offdiag - 1) + 1;
+
+ ## Column index
+ j = offdiag - (i - 1).*(i - 2)/2;
+
+ diagvals = randn (1, r);
+ offdiagvals = randn (1, m);
+
+ S = sparse ([ondiag, i, j], [ondiag, j, i],
+ [diagvals, offdiagvals, offdiagvals], n, n);
+
+endfunction
+
+function r = pick_rand_diag (n, k)
+ ## Pick a random number R of entries for the diagonal of a sparse NxN
+ ## symmetric square matrix with exactly K nonzero entries, ensuring
+ ## that this R is chosen uniformly over all such matrices.
+ ##
+ ## Let D be the number of diagonal entries and M the number of
+ ## off-diagonal entries. Then K = D + 2*M. Let A = N*(N-1)/2 be the
+ ## number of available entries in the upper triangle of the matrix.
+ ## Then, by a simple counting argument, there is a total of
+ ##
+ ## T = nchoosek (N, D) * nchoosek (A, M)
+ ##
+ ## symmetric NxN matrices with a total of K nonzero entries and D on
+ ## the diagonal. Letting D range from mod (K,2) through min (N,K), and
+ ## dividing by this sum, we obtain the probability P for D to be each
+ ## of those values.
+ ##
+ ## However, we cannot use this form for computation, as the binomial
+ ## coefficients become unmanageably large. Instead, we use the
+ ## successive quotients Q(i) = T(i+1)/T(i), which we easily compute to
+ ## be
+ ##
+ ## (N - D)*(N - D - 1)*M
+ ## Q = -------------------------------
+ ## (D + 2)*(D + 1)*(A - M + 1)
+ ##
+ ## Then, after prepending 1, the cumprod of these quotients is
+ ##
+ ## C = [ T(1)/T(1), T(2)/T(1), T(3)/T(1), ..., T(N)/T(1) ]
+ ##
+ ## Their sum is thus S = sum (T)/T(1), and then C(i)/S is the desired
+ ## probability P(i) for i=1:N. The cumsum will finally give the
+ ## distribution function for computing the random number of entries on
+ ## the diagonal R.
+ ##
+ ## Thanks to Zsbán Ambrus for most of the ideas
+ ## of the implementation here, especially how to do the computation
+ ## numerically to avoid overflow.
+
+ ## Degenerate case
+ if k == 1
+ r = 1;
+ return
+ endif
+
+ ## Compute the stuff described above
+ a = n*(n - 1)/2;
+ d = [mod(k,2):2:min(n,k)-2];
+ m = (k - d)/2;
+ q = (n - d).*(n - d - 1).*m ./ (d + 2)./(d + 1)./(a - m + 1);
+
+ ## Slight modification from discussion above: pivot around the max in
+ ## order to avoid overflow (underflow is fine, just means effectively
+ ## zero probabilities).
+ [~, midx] = max (cumsum (log (q))) ;
+ midx++;
+ lc = fliplr (cumprod (1./q(midx-1:-1:1)));
+ rc = cumprod (q(midx:end));
+
+ ## Now c = t(i)/t(midx), so c > 1 == [].
+ c = [lc, 1, rc];
+ s = sum (c);
+ p = c/s;
+
+ ## Add final d
+ d(end+1) = d(end) + 2;
+
+ ## Pick a random r using this distribution
+ r = d(sum (cumsum (p) < rand) + 1);
+
+endfunction
+
+%!test
+%! s = sprandsym (10, 0.1);
+%! assert (issparse (s));
+%! assert (issymmetric (s));
+%! assert (size (s), [10, 10]);
+%! assert (nnz (s) / numel (s), 0.1, .01);
+
+%% Test 1-input calling form
+%!test
+%! s = sprandsym (sparse ([1 2 3], [3 2 3], [2 2 2]));
+%! [i, j] = find (s);
+%! assert (sort (i), [2 3]');
+%! assert (sort (j), [2 3]');
+
+%% Test input validation
+%!error sprandsym ()
+%!error sprandsym (1, 2, 3)
+%!error sprandsym (ones(3), 0.5)
+%!error sprandsym (3.5, 0.5)
+%!error sprandsym (0, 0.5)
+%!error sprandsym (3, -1)
+%!error sprandsym (3, 2)
+