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OdeErrControl: Example and Test
Define X : R → R 2 by X 0 ( 0 ) = 1 X 1 ( 0 ) = 0 X 0 ( 1 ) ( t ) = - α X 0 ( t ) X 1 ( 1 ) ( t ) = 1 / X 0 ( t ) It follows that X 0 ( t ) = exp ( - α t ) X 1 ( t ) = [ exp ( α t ) - 1 ] / α This example tests OdeErrControl using the relations above.
Nan
Note that X 0 ( t ) > 0 for all t and that the ODE goes through a singularity between X 0 ( t ) > 0 and X 0 ( t ) < 0 . If X 0 ( t ) < 0 , we return `nan` in order to inform `OdeErrControl` that its is taking to large a step. `

    # include <limits>                      // for quiet_NaN
    # 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:
              const double alpha_;
         public:
              // constructor
              Fun(double alpha) : alpha_(alpha)
              { }

              // set f = x'(t)
              void Ode(
                   const double                &t,
                   const CppAD::vector<double> &x,
                   CppAD::vector<double>       &f)
              {     f[0] = - alpha_ * x[0];
                   f[1] = 1. / x[0];
                   // case where ODE does not make sense
                   if( x[0] < 0. )
                        f[1] = std::numeric_limits<double>::quiet_NaN();
              }

         };

         // --------------------------------------------------------------
         class Method {
         private:
              Fun F;
         public:
              // constructor
              Method(double alpha) : F(alpha)
              { }
              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 OdeErrControl(void)
    {     bool ok = true;     // initial return value

         double alpha = 10.;
         Method method(alpha);

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

         CppAD::vector<double> eabs(2);
         eabs[0] = 1e-4;
         eabs[1] = 1e-4;

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

         // outputs
         CppAD::vector<double> ef(2);
         CppAD::vector<double> xf(2);
         CppAD::vector<double> maxabs(2);
         size_t nstep;

         xf = OdeErrControl(method,
              ti, tf, xi, smin, smax, scur, eabs, erel, ef, maxabs, nstep);

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

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

         return ok;
    }

`

Input File: example/utility/ode_err_control.cpp