Runge45: Example and Test

runge45_1.cpp

Headings

@(@\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}} }@)@ 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{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} @]@ 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;
}

Input File: example/utility/runge45_1.cpp