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X : \R \rightarrow \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/ode_gear_control.hpp> // CppAD::OdeGearControl
namespace {
// --------------------------------------------------------------
class Fun {
private:
CPPAD_TEST_VECTOR<double> w;
public:
// constructor
Fun(const CPPAD_TEST_VECTOR<double> &w_) : w(w_)
{ }
// set f = x'(t)
template <typename Scalar>
void Ode(
const Scalar &t,
const CPPAD_TEST_VECTOR<Scalar> &x,
CPPAD_TEST_VECTOR<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_TEST_VECTOR<double> &x,
CPPAD_TEST_VECTOR<double> &f_x)
{ using namespace CppAD;
size_t n = x.size();
CPPAD_TEST_VECTOR< AD<double> > T(1);
CPPAD_TEST_VECTOR< AD<double> > X(n);
CPPAD_TEST_VECTOR< 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_TEST_VECTOR<double> dx(n);
CPPAD_TEST_VECTOR<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
CPPAD_TEST_VECTOR<double> w(2);
w[0] = 10.;
w[1] = 1.;
Fun F(w);
CPPAD_TEST_VECTOR<double> xi(2);
xi[0] = 1.;
xi[1] = 0.;
CPPAD_TEST_VECTOR<double> eabs(2);
eabs[0] = 1e-4;
eabs[1] = 1e-4;
// return values
CPPAD_TEST_VECTOR<double> ef(2);
CPPAD_TEST_VECTOR<double> maxabs(2);
CPPAD_TEST_VECTOR<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 = 1e-10;
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 &= CppAD::NearEqual(x0, xf[0], 1e-4, 1e-4);
ok &= CppAD::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 &= CppAD::NearEqual(x1, xf[1], 1e-4, 1e-4);
ok &= CppAD::NearEqual(0., ef[1], 1e-4, 1e-4);
return ok;
}