# 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_TESTVECTOR(Scalar) &x,
CPPAD_TESTVECTOR(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_TESTVECTOR(double) &x,
CPPAD_TESTVECTOR(double) &f_t)
{ 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;
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_TESTVECTOR(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_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.;
}
}
private:
const bool use_x;
};
}
bool Rosen34(void)
{ bool ok = true; // initial return value
size_t i; // temporary indices
using CppAD::NearEqual;
double eps99 = 99.0 * std::numeric_limits<double>::epsilon();
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_TESTVECTOR(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_TESTVECTOR(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 &=
NearEqual(xf[i], check, eps99, eps99);
// 3rd order method is exact for i < 3
if( i < 3 )
ok &= (e[i] <= eps99);
// check value for next i
check *= tf;
}
}
return ok;
}