ode_gear_control.cpp¶

OdeGearControl: Example and Test¶

Define \(X : \B{R} \rightarrow \B{R}^2\) by

\begin{eqnarray} X_0 (t) & = & - \exp ( - w_0 t ) \\ X_1 (t) & = & \frac{w_0}{w_1 - w_0} [ \exp ( - w_0 t ) - \exp( - w_1 t )] \end{eqnarray}

It follows that \(X_0 (0) = 1\), \(X_1 (0) = 0\) and

\begin{eqnarray} X_0^{(1)} (t) & = & - w_0 X_0 (t) \\ X_1^{(1)} (t) & = & + w_0 X_0 (t) - w_1 X_1 (t) \end{eqnarray}

The example tests OdeGearControl using the relations above:

# include <cppad/cppad.hpp>
# include <cppad/utility/ode_gear_control.hpp>   // CppAD::OdeGearControl

namespace {
   // --------------------------------------------------------------
   class Fun {
   private:
         CPPAD_TESTVECTOR(double) w;
   public:
      // constructor
      Fun(const CPPAD_TESTVECTOR(double) &w_) : w(w_)
      { }

      // set f = x'(t)
      template <class Scalar>
      void Ode(
         const Scalar                    &t,
         const CPPAD_TESTVECTOR(Scalar) &x,
         CPPAD_TESTVECTOR(Scalar)       &f)
      {  f[0] = - w[0] * x[0];
         f[1] = + w[0] * x[0] - w[1] * x[1];
      }

      void Ode_dep(
         const double                    &t,
         const CPPAD_TESTVECTOR(double) &x,
         CPPAD_TESTVECTOR(double)       &f_x)
      {  using namespace CppAD;

         size_t n  = x.size();
         CPPAD_TESTVECTOR(AD<double>) T(1);
         CPPAD_TESTVECTOR(AD<double>) X(n);
         CPPAD_TESTVECTOR(AD<double>) F(n);

         // set argument values
         T[0] = t;
         size_t i, j;
         for(i = 0; i < n; i++)
            X[i] = x[i];

         // declare independent variables
         Independent(X);

         // compute f(t, x)
         this->Ode(T[0], X, F);

         // define AD function object
         ADFun<double> fun(X, F);

         // compute partial of f w.r.t x
         CPPAD_TESTVECTOR(double) dx(n);
         CPPAD_TESTVECTOR(double) df(n);
         for(j = 0; j < n; j++)
            dx[j] = 0.;
         for(j = 0; j < n; j++)
         {  dx[j] = 1.;
            df = fun.Forward(1, dx);
            for(i = 0; i < n; i++)
               f_x [i * n + j] = df[i];
            dx[j] = 0.;
         }
      }
   };
}

bool OdeGearControl(void)
{  bool ok = true;     // initial return value
   using CppAD::NearEqual;
   double eps99 = 99.0 * std::numeric_limits<double>::epsilon();

   CPPAD_TESTVECTOR(double) w(2);
   w[0] = 10.;
   w[1] = 1.;
   Fun F(w);

   CPPAD_TESTVECTOR(double) xi(2);
   xi[0] = 1.;
   xi[1] = 0.;

   CPPAD_TESTVECTOR(double) eabs(2);
   eabs[0] = 1e-4;
   eabs[1] = 1e-4;

   // return values
   CPPAD_TESTVECTOR(double) ef(2);
   CPPAD_TESTVECTOR(double) maxabs(2);
   CPPAD_TESTVECTOR(double) xf(2);
   size_t                nstep;

   // input values
   size_t  M   = 5;
   double ti   = 0.;
   double tf   = 1.;
   double smin = 1e-8;
   double smax = 1.;
   double sini = eps99;
   double erel = 0.;

   xf = CppAD::OdeGearControl(F, M,
      ti, tf, xi, smin, smax, sini, eabs, erel, ef, maxabs, nstep);

   double x0 = exp(-w[0]*tf);
   ok &= NearEqual(x0, xf[0], 1e-4, 1e-4);
   ok &= NearEqual(0., ef[0], 1e-4, 1e-4);

   double x1 = w[0] * (exp(-w[0]*tf) - exp(-w[1]*tf))/(w[1] - w[0]);
   ok &= NearEqual(x1, xf[1], 1e-4, 1e-4);
   ok &= NearEqual(0., ef[1], 1e-4, 1e-4);

   return ok;
}