On 11/24/2012 2:51 PM, Robert Dodier wrote:
> We could obviate the lack of documentation by implementing something
> more sophisticated than Romberg's method for bigfloats
It may be worth pointing out that methods in the typical numerical
library (like quadpack), as well as romberg, operate with functions that
are implemented as black boxes. That is, the program is given some
f(x) as a program, and all it can do is call f on specific numeric values.
Some numerical programs are more appropriate with some kinds of
functions -- but it is generally the human user who has to appreciate
the situation
and choose the right program. Though see
http://dl.acm.org/citation.cfm?id=321870
or what NIntegrate in Mathematica does.
By contrast, A sophisticated numerical integration program in a computer
algebra
system can consider the case where f is known as a formula, and
therefore it can be analyzed in various ways -- search for singularities,
asymptotes, etc. It may be possible to expand in a Taylor series, or
approximate as a rational function. A neat trick appropriate for a certain
broad class of oscillatory integrands is to expand in a series by
integration by parts.
If you