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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}} }@)@ This is cppad-20221105 documentation. Here is a link to its current documentation .
OdeErrControl: Example and Test
Define @(@ X : \B{R} \rightarrow \B{R}^2 @)@ by @[@ \begin{array}{rcl} X_0 (0) & = & 1 \\ X_1 (0) & = & 0 \\ X_0^{(1)} (t) & = & - \alpha X_0 (t) \\ X_1^{(1)} (t) & = & 1 / X_0 (t) \end{array} @]@ It follows that @[@ \begin{array}{rcl} X_0 (t) & = & \exp ( - \alpha t ) \\ X_1 (t) & = & [ \exp( \alpha t ) - 1 ] / \alpha \end{array} @]@ 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