OdeGearControl: Example and Test

ode_gear_control.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}} }@)@ OdeGearControl: Example and Test 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} @]@ The example tests OdeGearControl using the relations above:



# include <cppad/cppad.hpp>
# include <cppad/utility/ode_gear_control.hpp>   // CppAD::OdeGearControl

namespace {
     // --------------------------------------------------------------
     class Fun {
     private:
           CPPAD_TESTVECTOR(double) w;
     public:
          // constructor
          Fun(const CPPAD_TESTVECTOR(double) &w_) : w(w_)
          { }

          // set f = x'(t)
          template <typename Scalar>
          void Ode(
               const Scalar                    &t,
               const CPPAD_TESTVECTOR(Scalar) &x,
               CPPAD_TESTVECTOR(Scalar)       &f)
          {     f[0] = - w[0] * x[0];
               f[1] = + w[0] * x[0] - w[1] * x[1];
          }

          void Ode_dep(
               const double                    &t,
               const CPPAD_TESTVECTOR(double) &x,
               CPPAD_TESTVECTOR(double)       &f_x)
          {     using namespace CppAD;

               size_t n  = x.size();
               CPPAD_TESTVECTOR(AD<double>) T(1);
               CPPAD_TESTVECTOR(AD<double>) X(n);
               CPPAD_TESTVECTOR(AD<double>) F(n);

               // set argument values
               T[0] = t;
               size_t i, j;
               for(i = 0; i < n; i++)
                    X[i] = x[i];

               // declare independent variables
               Independent(X);

               // compute f(t, x)
               this->Ode(T[0], X, F);

               // define AD function object
               ADFun<double> fun(X, F);

               // compute partial of f w.r.t x
               CPPAD_TESTVECTOR(double) dx(n);
               CPPAD_TESTVECTOR(double) df(n);
               for(j = 0; j < n; j++)
                    dx[j] = 0.;
               for(j = 0; j < n; j++)
               {     dx[j] = 1.;
                    df = fun.Forward(1, dx);
                    for(i = 0; i < n; i++)
                         f_x [i * n + j] = df[i];
                    dx[j] = 0.;
               }
          }
     };
}

bool OdeGearControl(void)
{     bool ok = true;     // initial return value
     using CppAD::NearEqual;
     double eps99 = 99.0 * std::numeric_limits<double>::epsilon();

     CPPAD_TESTVECTOR(double) w(2);
     w[0] = 10.;
     w[1] = 1.;
     Fun F(w);

     CPPAD_TESTVECTOR(double) xi(2);
     xi[0] = 1.;
     xi[1] = 0.;

     CPPAD_TESTVECTOR(double) eabs(2);
     eabs[0] = 1e-4;
     eabs[1] = 1e-4;

     // return values
     CPPAD_TESTVECTOR(double) ef(2);
     CPPAD_TESTVECTOR(double) maxabs(2);
     CPPAD_TESTVECTOR(double) xf(2);
     size_t                nstep;

     // input values
     size_t  M   = 5;
     double ti   = 0.;
     double tf   = 1.;
     double smin = 1e-8;
     double smax = 1.;
     double sini = eps99;
     double erel = 0.;

     xf = CppAD::OdeGearControl(F, M,
          ti, tf, xi, smin, smax, sini, eabs, erel, ef, maxabs, nstep);

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

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

     return ok;
}

Input File: example/utility/ode_gear_control.cpp