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ode_err_maxabs.cpp

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OdeErrControl: Example and Test Using Maxabs Argument

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}

Note that \(X_1 (0)\) is zero an if \(w_0 t\) is large, \(X_0 (t)\) is near zero. This example tests OdeErrControl using the maxabs argument.

# include <cstddef>              // for size_t
# include <cmath>                // for exp
# include <cppad/utility/ode_err_control.hpp>   // CppAD::OdeErrControl
# include <cppad/utility/near_equal.hpp>    // CppAD::NearEqual
# include <cppad/utility/vector.hpp> // CppAD::vector
# include <cppad/utility/runge_45.hpp>      // CppAD::Runge45

namespace {
   // --------------------------------------------------------------
   class Fun {
   private:
         CppAD::vector<double> w;
   public:
      // constructor
      Fun(const CppAD::vector<double> &w_) : w(w_)
      { }

      // set f = x'(t)
      void Ode(
         const double                &t,
         const CppAD::vector<double> &x,
         CppAD::vector<double>       &f)
      {  f[0] = - w[0] * x[0];
         f[1] = + w[0] * x[0] - w[1] * x[1];
      }
   };

   // --------------------------------------------------------------
   class Method {
   private:
      Fun F;
   public:
      // constructor
      Method(const CppAD::vector<double> &w_) : F(w_)
      { }
      void step(
         double ta,
         double tb,
         CppAD::vector<double> &xa ,
         CppAD::vector<double> &xb ,
         CppAD::vector<double> &eb )
      {  xb = CppAD::Runge45(F, 1, ta, tb, xa, eb);
      }
      size_t order(void)
      {  return 4; }
   };
}

bool OdeErrMaxabs(void)
{  bool ok = true;     // initial return value

   CppAD::vector<double> w(2);
   w[0] = 10.;
   w[1] = 1.;
   Method method(w);

   CppAD::vector<double> xi(2);
   xi[0] = 1.;
   xi[1] = 0.;

   CppAD::vector<double> eabs(2);
   eabs[0] = 0.;
   eabs[1] = 0.;

   CppAD::vector<double> ef(2);
   CppAD::vector<double> xf(2);
   CppAD::vector<double> maxabs(2);

   double ti   = 0.;
   double tf   = 1.;
   double smin = .5;
   double smax = 1.;
   double scur = .5;
   double erel = 1e-4;

   bool accurate = false;
   while( ! accurate )
   {  xf = OdeErrControl(method,
         ti, tf, xi, smin, smax, scur, eabs, erel, ef, maxabs);
      accurate = true;
      size_t i;
      for(i = 0; i < 2; i++)
         accurate &= ef[i] <= erel * maxabs[i];
      if( ! accurate )
         smin = smin / 2;
   }

   double x0 = exp(-w[0]*tf);
   ok &= CppAD::NearEqual(x0, xf[0], erel, 0.);
   ok &= CppAD::NearEqual(0., ef[0], erel, erel);

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

   return ok;
}