[GiNaC-devel] Numerical integration in GiNaC

[Chris Dams]

Dear Richy,

On Mon, 2005-09-12 at 22:16 +0200, Richard B. Kreckel wrote:

> 1) Why the decision to *not* couple the precision to the value of Digits?
>    Coupling it to Digits would make it available in Ginsh.  Wouldn't it do
>    the right thing in almost all cases?

It depends on what coupling you have in mind. As different users may
have different preferences, I thought it was best to leave it up to the
user. Note that it would be a highly dangerous idea to make the
precision of integration equal to the precision of the other numerical
calculations. This is because the last few digits of a calculated number
may not be very trustworthy and all kind of artifacts could be in them.
A continuous function could look non-continuous depending on the
algorithm used for calculation. Then the adaptive simpson method would
go *boink*.

> 2) Are variations of Simpson's rule really attractive for numerical
>    evaluation?  Have you considered/tried something funkier (maybe
>    Romberg's method) or is that a particularly smart implementation?

I thought it was quite nice. Convergence as 1/N^4 and adaptivity. It has
been some time that I looked at this, so I forgot why I took the
adaptive Simpson algorithm. Presumably there was something in the book
by Burden and Faires that made me go for the adaptive Simpson algorithm.
Unfortunately I do not have that book now that I am in Milan and the
library seems to be reorganizing until december or so :-(.

Of course, there could also be an integral::integration_method to give
the user a choice what method to use.

> 3) Probably an oversight: the last argument to the adaptivesimpson
>    function is unused.  Instead, the code always uses the default.

Yes, the obvious fix for this is to turn all occurances of
integral::relative_integration_error into error. Could you do this in
CVS?

Best wishes,
Chris