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X : \R \rightarrow \R^n
by
\[
X_i (t) = t^{i+1}
\]
for
i = 1 , \ldots , n-1
.
It follows that
\[
\begin{array}{rclr}
X_i(0) & = & 0 & {\rm for \; all \;} i \\
X_i ' (t) & = & 1 & {\rm if \;} i = 0 \\
X_i '(t) & = & (i+1) t^i = (i+1) X_{i-1} (t) & {\rm if \;} i > 0
\end{array}
\]
The example tests Rosen34 using the relations above:
# include <cppad/cppad.hpp> // For automatic differentiation
namespace {
class Fun {
public:
// constructor
Fun(bool use_x_) : use_x(use_x_)
{ }
// compute f(t, x) both for double and AD<double>
template <typename Scalar>
void Ode(
const Scalar &t,
const CPPAD_TEST_VECTOR<Scalar> &x,
CPPAD_TEST_VECTOR<Scalar> &f)
{ size_t n = x.size();
Scalar ti(1);
f[0] = Scalar(1);
size_t i;
for(i = 1; i < n; i++)
{ ti *= t;
// convert int(size_t) to avoid warning
// on _MSC_VER systems
if( use_x )
f[i] = int(i+1) * x[i-1];
else f[i] = int(i+1) * ti;
}
}
// compute partial of f(t, x) w.r.t. t using AD
void Ode_ind(
const double &t,
const CPPAD_TEST_VECTOR<double> &x,
CPPAD_TEST_VECTOR<double> &f_t)
{ 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;
for(i = 0; i < n; i++)
X[i] = x[i];
// declare independent variables
Independent(T);
// compute f(t, x)
this->Ode(T[0], X, F);
// define AD function object
ADFun<double> Fun(T, F);
// compute partial of f w.r.t t
CPPAD_TEST_VECTOR<double> dt(1);
dt[0] = 1.;
f_t = Fun.Forward(1, dt);
}
// compute partial of f(t, x) w.r.t. x using AD
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.;
}
}
private:
const bool use_x;
};
}
bool Rosen34(void)
{ bool ok = true; // initial return value
size_t i; // temporary indices
size_t n = 4; // number components in X(t) and order of method
size_t M = 2; // number of Rosen34 steps in [ti, tf]
double ti = 0.; // initial time
double tf = 2.; // final time
// xi = X(0)
CPPAD_TEST_VECTOR<double> xi(n);
for(i = 0; i <n; i++)
xi[i] = 0.;
size_t use_x;
for( use_x = 0; use_x < 2; use_x++)
{ // function object depends on value of use_x
Fun F(use_x > 0);
// compute Rosen34 approximation for X(tf)
CPPAD_TEST_VECTOR<double> xf(n), e(n);
xf = CppAD::Rosen34(F, M, ti, tf, xi, e);
double check = tf;
for(i = 0; i < n; i++)
{ // check that error is always positive
ok &= (e[i] >= 0.);
// 4th order method is exact for i < 4
if( i < 4 ) ok &=
CppAD::NearEqual(xf[i], check, 1e-10, 1e-10);
// 3rd order method is exact for i < 3
if( i < 3 )
ok &= (e[i] <= 1e-10);
// check value for next i
check *= tf;
}
}
return ok;
}