C++ for Scientists - Technische Universität Dresden

116 CHAPTER 4. GENERIC PROGRAMMING
template
class second derivative
{
public:
second derivative(const F& f, const T& h) : h(h), fp(f, h) {}
T operator()(const T& x) const
{
return ( fp(x+h) − fp(x) ) / h ;
}
private:
T h;
derivative fp;
};
Now we can build the f ′′ functor from f:
second derivative spc scd2(para sin plus cos(1.0), 0.001);
When we think about how we would implement the third, fourth or in general the n-th derivative,
we realize that they would look much like the second one: calling the (n-1)-th derivative on x+h
and x. We can explore this with a recursive implementation:
template
class nth derivative
{
typedef nth derivative prec derivative;
public:
nth derivative(const F& f, const T& h) : h(h), fp(f, h) {}
T operator()(const T& x) const
{
return ( fp(x+h) − fp(x) ) / h ;
}
private:
T h;
prec derivative fp;
};
To save the compiler from infinite recursion we must stop this mutual referring when we reach
the first derivative. Note that we cannot use ‘if’ or ‘?:’ to stop the recursion because both of its
respective branches are evaluated and one of them still contains the infinite recursion. Recursive
template definitions are terminated with a specialization like this:
template
class nth derivative
{
public:
nth derivative(const F& f, const T& h) : f(f), h(h) {}
T operator()(const T& x) const
{
return ( f(x+h) − f(x) ) / h ;
}
private: