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@(@\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@
ColPack: Sparse Jacobian Example and Test

# include <cppad/cppad.hpp>
bool colpack_jac(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;

     // domain space vector
     size_t n = 4;
     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);

     size_t m = 3;
     a_vector  a_y(m);
     a_y[0] = a_x[0] + a_x[1];
     a_y[1] = a_x[2] + a_x[3];
     a_y[2] = a_x[0] + a_x[1] + a_x[2] + a_x[3] * a_x[3] / 2.;

     // 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 1 0 0  ]
     jac = [ 0 0 1 1  ]
           [ 1 1 1 x_3]
     */
     // Normally one would use CppAD to compute sparsity pattern, but for this
     // example we set it directly
     size_t nr  = m;
     size_t nc  = n;
     size_t nnz = 8;
     sparsity pattern(nr, nc, nnz);
     d_vector check(nnz);
     for(size_t k = 0; k < nnz; k++)
     {     size_t r, c;
          if( k < 2 )
          {     r = 0;
               c = k;
          }
          else if( k < 4 )
          {     r = 1;
               c = k;
          }
          else
          {     r = 2;
               c = k - 4;
          }
          pattern.set(k, r, c);
          if( k == nnz - 1 )
               check[k] = x[3];
          else
               check[k] = 1.0;
     }

     // using row and column indices to compute non-zero in rows 1 and 2
     sparse_matrix subset( pattern );

     // check results for both CppAD and Colpack
     for(size_t i_method = 0; i_method < 4; i_method++)
     {     // coloring method
          std::string coloring;
          if( i_method % 2 == 0 )
               coloring = "cppad";
          else
               coloring = "colpack";
          //
          CppAD::sparse_jac_work work;
          size_t group_max = 1;
          if( i_method / 2 == 0 )
          {     size_t n_sweep = f.sparse_jac_for(
                    group_max, x, subset, pattern, coloring, work
               );
               ok &= n_sweep == 4;
          }
          else
          {     size_t n_sweep = f.sparse_jac_rev(
                    x, subset, pattern, coloring, work
               );
               ok &= n_sweep == 2;
          }
          const d_vector& hes( subset.val() );
          for(size_t k = 0; k < nnz; k++)
               ok &= check[k] == hes[k];
     }
     return ok;
}

Input File: example/sparse/colpack_jac.cpp