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Correctness Check for Both Simple and Fast Representations
`

    # include "ode_run.hpp"

    bool ode_check(const SizeVector& N, const NumberVector& x)
    {     bool ok = true;
         size_t i, j;

         // number of components of x corresponding to values for y
         size_t ny_inx = x.size() - Na;

         // compute the partial sums of the number of grid points
         // and the maximum step size for the trapezoidal approximation
         SizeVector S(Nz+1);
         S[0] = 0;
         Number max_step = 0.;
         for(i = 1; i <= Nz; i++)
         {     S[i] = S[i-1] + N[i];
              max_step = std::max(max_step, Number(s[i] - s[i-1]) / Number(N[i]) );
         }

         // split out return values
         NumberVector a(Na), y_0(Ny), y_1(Ny), y_2(Ny);
         for(j = 0; j < Na; j++)
              a[j] = x[ny_inx+j];
         for(j = 0; j < Ny; j++)
         {     y_0[j] = x[j];
              y_1[j] = x[Ny + j];
              y_2[j] = x[2 * Ny + j];
         }

         // Check some of the optimal a value
         Number rel_tol = max_step * max_step;
         Number abs_tol = rel_tol;
         Number check_a[] = {a0, a1, a2}; // see the y_one function
         for(j = 0; j < Na; j++)
         {
              ok &= CppAD::NearEqual(
                   check_a[j], a[j], rel_tol, abs_tol
              );
         }

         // check accuarcy of constraint equations
         rel_tol = 1e-9;
         abs_tol = 1e-9;

         // check the initial value constraint
         NumberVector F = eval_F(a);
         for(j = 0; j < Ny; j++)
              ok &= CppAD::NearEqual(F[j], y_0[j], rel_tol, abs_tol);

         // check the first trapezoidal equation
         NumberVector G_0 = eval_G(y_0, a);
         NumberVector G_1 = eval_G(y_1, a);
         Number dt = (s[1] - s[0]) / Number(N[1]);
         Number check;
         for(j = 0; j < Ny; j++)
         {     check = y_1[j] - y_0[j] - (G_1[j]+G_0[j])*dt/2;
              ok &= CppAD::NearEqual( check, 0., rel_tol, abs_tol);
         }
         //
         // check the second trapezoidal equation
         NumberVector G_2 = eval_G(y_2, a);
         if( N[1] == 1 )
              dt = (s[2] - s[1]) / Number(N[2]);
         for(j = 0; j < Ny; j++)
         {     check = y_2[j] - y_1[j] - (G_2[j]+G_1[j])*dt/2;
              ok &= CppAD::NearEqual( check, 0., rel_tol, abs_tol);
         }
         //
         // check the objective function (specialized to this case)
         check = 0.;
         NumberVector y_i(Ny);
         for(size_t k = 0; k < Nz; k++)
         {     for(j = 0; j < Ny; j++)
                   y_i[j] =  x[S[k+1] * Ny + j];
              check += eval_H<Number>(k + 1, y_i, a);
         }
         Number obj_value = 0.; // optimal object (no noise in simulation)
         ok &= CppAD::NearEqual(check, obj_value, rel_tol, abs_tol);

         // Use this empty namespace function to avoid warning that it is not used
         static size_t ode_check_count = 0;
         ode_check_count++;
         ok &= count_eval_r() == ode_check_count;

         return ok;
    }

`

Input File: cppad_ipopt/example/ode_check.cpp