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@(@\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}} }@)@
OdeErrControl: Example and Test Using Maxabs Argument
Define @(@ X : \B{R} \rightarrow \B{R}^2 @)@ by @[@ \begin{array}{rcl} 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{array} @]@ It follows that @(@ X_0 (0) = 1 @)@, @(@ X_1 (0) = 0 @)@ and @[@ \begin{array}{rcl} X_0^{(1)} (t) & = & - w_0 X_0 (t) \\ X_1^{(1)} (t) & = & + w_0 X_0 (t) - w_1 X_1 (t) \end{array} @]@ 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;
}

Input File: example/utility/ode_err_maxabs.cpp