[boost] Thoughts on Numeric Integration

From: Daryle Walker (dwalker07_at_[hidden])
Date: 2002-10-24 03:26:45

For now, let's stick with functions that map one real number to another 
(so no complex numbers or vector mapping(s)).  A basic function design 
could be:

template < typename ResultType, typename DomainType, typename RangeType 
 >
ResultType
integrate
(
     std::map<DomainType, RangeType>  m,
     ResultType                       initial_condition = ResultType()
);

This is pure numerical integration without assuming any distribution 
pattern for the domain or range values (except that all the domain 
values are unique).

That function can't work unless all the points are known in advance.  
If we build the graph as we go, we need something like:

template < typename ResultType, typename DomainType, typename RangeType 
 >
class integrator
{
public:
     explicit  integrator( ResultType initial_condition = ResultType() );

     void  update( DomainType x, RangeType y );

     ResultType  current_result() const;
};

We should insist that updates keep new "x" values strictly increasing 
(or risk undefined behavior), so we don't have to keep an arbitrary 
history.

Now for the variants:

A.	The domain values can be given be an STL iterator sequence; an 
initial value, step size, and final value or step count; a generator 
(C++) function/functor.

B.	The range values can be given in similar ways, or with a (C++) 
function/functor that takes processes the current domain value.  The 
latter option could be combined with hints about the (math) function's 
discontinuities or undefined regions.

C.	The domain and range values can be given in a combined STL iterator 
sequence of tuples.

Note that specific techniques are not mentioned.  If there are several 
ways to do an integration (rectangular rule, trapezoidal rule, Simpon's 
Rule, etc.), we can provide similar (C++) functions or classes with the 
same signature, or policies could be used.  (I don't know too much 
about the "policy" philosophy with C++ programming techniques, could 
someone explain it to me privately?)

Daryle