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runge45_1.cpp

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Runge45: Example and Test

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

\[X_i (t) = t^{i+1}\]

for \(i = 1 , \ldots , n-1\). It follows that

\[\begin{split}\begin{array}{rclr} X_i(0) & = & 0 & {\rm for \; all \;} i \\ X_i ' (t) & = & 1 & {\rm if \;} i = 0 \\ X_i '(t) & = & (i+1) t^i = (i+1) X_{i-1} (t) & {\rm if \;} i > 0 \end{array}\end{split}\]

The example tests Runge45 using the relations above:

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

// Runge45 requires fabs to be defined (not std::fabs)
// <cppad/cppad.hpp> defines this for doubles, but runge_45.hpp does not.
# include <math.h>      // for fabs without std in front

namespace {
   class Fun {
   public:
      // constructor
      Fun(bool use_x_) : use_x(use_x_)
      { }

      // set f = x'(t)
      void Ode(
         const double                &t,
         const CppAD::vector<double> &x,
         CppAD::vector<double>       &f)
      {  size_t n  = x.size();
         double ti = 1.;
         f[0]      = 1.;
         size_t i;
         for(i = 1; i < n; i++)
         {  ti *= t;
            if( use_x )
               f[i] = double(i+1) * x[i-1];
            else
               f[i] = double(i+1) * ti;
         }
      }
   private:
      const bool use_x;

   };
}

bool runge_45_1(void)
{  bool ok = true;     // initial return value
   size_t i;           // temporary indices

   using CppAD::NearEqual;
   double eps99 = 99.0 * std::numeric_limits<double>::epsilon();

   size_t  n = 5;      // number components in X(t) and order of method
   size_t  M = 2;      // number of Runge45 steps in [ti, tf]
   double ti = 0.;     // initial time
   double tf = 2.;     // final time

   // xi = X(0)
   CppAD::vector<double> xi(n);
   for(i = 0; i <n; i++)
      xi[i] = 0.;

   size_t use_x;
   for( use_x = 0; use_x < 2; use_x++)
   {  // function object depends on value of use_x
      Fun F(use_x > 0);

      // compute Runge45 approximation for X(tf)
      CppAD::vector<double> xf(n), e(n);
      xf = CppAD::Runge45(F, M, ti, tf, xi, e);

      double check = tf;
      for(i = 0; i < n; i++)
      {  // check that error is always positive
         ok    &= (e[i] >= 0.);
         // 5th order method is exact for i < 5
         if( i < 5 ) ok &=
            NearEqual(xf[i], check, eps99, eps99);
         // 4th order method is exact for i < 4
         if( i < 4 )
            ok &= (e[i] <= eps99);

         // check value for next i
         check *= tf;
      }
   }
   return ok;
}