# include <cppad/cppad.hpp>
bool colpack_hes(void)
{     bool ok = true;
     using CppAD::AD;
     using CppAD::NearEqual;
     typedef CPPAD_TESTVECTOR(AD<double>)            a_vector;
     typedef CPPAD_TESTVECTOR(double)                d_vector;
     typedef CppAD::vector<size_t>                   i_vector;
     typedef CppAD::sparse_rc<i_vector>              sparsity;
     typedef CppAD::sparse_rcv<i_vector, d_vector>   sparse_matrix;
     double eps = 10. * CppAD::numeric_limits<double>::epsilon();
     //
     // domain space vector
     size_t n = 5;
     a_vector  a_x(n);
     for(size_t j = 0; j < n; j++)
          a_x[j] = AD<double> (0);
     //
     // declare independent variables and starting recording
     CppAD::Independent(a_x);

     // colpack example case where hessian is a spear head
     // i.e, H(i, j) non zero implies i = 0, j = 0, or i = j
     AD<double> sum = 0.0;
     // partial_0 partial_j = x[j]
     // partial_j partial_j = x[0]
     for(size_t j = 1; j < n; j++)
          sum += a_x[0] * a_x[j] * a_x[j] / 2.0;
     //
     // partial_i partial_i = 2 * x[i]
     for(size_t i = 0; i < n; i++)
          sum += a_x[i] * a_x[i] * a_x[i] / 3.0;

     // declare dependent variables
     size_t m = 1;
     a_vector  a_y(m);
     a_y[0] = sum;

     // create f: x -> y and stop tape recording
     CppAD::ADFun<double> f(a_x, a_y);

     // new value for the independent variable vector
     d_vector x(n);
     for(size_t j = 0; j < n; j++)
          x[j] = double(j + 1);

     /*
           [ 2  2  3  4  5 ]
     hes = [ 2  5  0  0  0 ]
           [ 3  0  7  0  0 ]
           [ 4  0  0  9  0 ]
           [ 5  0  0  0 11 ]
     */
     // Normally one would use CppAD to compute sparsity pattern, but for this
     // example we set it directly
     size_t nr  = n;
     size_t nc  = n;
     size_t nnz = n + 2 * (n - 1);
     sparsity pattern(nr, nc, nnz);
     for(size_t k = 0; k < n; k++)
     {     size_t r = k;
          size_t c = k;
          pattern.set(k, r, c);
     }
     for(size_t i = 1; i < n; i++)
     {     size_t k = n + 2 * (i - 1);
          size_t r = i;
          size_t c = 0;
          pattern.set(k,   r, c);
          pattern.set(k+1, c, r);
     }

     // subset of elements to compute
     // (only compute lower traingle)
     nnz = n + (n - 1);
     sparsity lower_triangle(nr, nc, nnz);
     d_vector check(nnz);
     for(size_t k = 0; k < n; k++)
     {     size_t r = k;
          size_t c = k;
          lower_triangle.set(k, r, c);
          check[k] = 2.0 * x[k];
          if( k > 0 )
               check[k] += x[0];
     }
     for(size_t j = 1; j < n; j++)
     {     size_t k = n + (j - 1);
          size_t r = 0;
          size_t c = j;
          lower_triangle.set(k, r, c);
          check[k] = x[c];
     }
     sparse_matrix subset( lower_triangle );

     // check results for both CppAD and Colpack
     for(size_t i_method = 0; i_method < 4; i_method++)
     {     // coloring method
          std::string coloring;
          switch(i_method)
          {     case 0:
               coloring = "cppad.symmetric";
               break;

               case 1:
               coloring = "cppad.general";
               break;

               case 2:
               coloring = "colpack.symmetric";
               break;

               case 3:
               coloring = "colpack.general";
               break;
          }
          //
          // compute Hessian
          CppAD::sparse_hes_work work;
          d_vector w(m);
          w[0] = 1.0;
          size_t n_sweep = f.sparse_hes(
               x, w, subset, pattern, coloring, work
          );
          //
          // check result
          const d_vector& hes( subset.val() );
          for(size_t k = 0; k < nnz; k++)
               ok &= NearEqual(check[k], hes[k], eps, eps);
          if(
               coloring == "cppad.symmetric"
          ||     coloring == "colpack.symmetric"
          )
               ok &= n_sweep == 2;
          else
               ok &= n_sweep == 5;
     }

     return ok;
}