On Sun, Apr 19, 2009 at 7:44 AM, Maurizio <maurizio.gran...@gmail.com> wrote:
> Carl, I took advantage of your suggestion, even though I assume I
> can't still go through the whole process with the current gcd
> capabilities in Pynac. But before than that, I'd like to point out
> something strange I did notice, and maybe also Burcin can help with
> that:
>
> reset()
> P.<x,z> = QQ[]
>
> B = x^3 + x
>
> var('x, zs', ns = 1)
> from sage.symbolic.ring import NSR
> Bs = NSR(B)
> Bs
> x^3 + x
> Bs.diff(x)
> 0
>
> So, the derivative is not working. Which is the cause? It seems that
> the "x" in Bs is not the "x" I declared, so the derivative gets 0 as a
> result. Which is the reason?
Looks like a bug to me.
Burcin, any comments?
> Moreover, I don't see this being the right way to do this, because
> (for this particular problem: integration) I don't like having the
> numerical representation of things like sqrt(5), even if the result is
> still correct, so that
>
> temp = QQbar(sqrt(5)); temp
> 2.236067977499790?
>
> temp^2
> 5.000000000000000?
Note that it's only the representation that's numerical; internally
these are actually exact numbers. (For instance, if you do "temp^2 ==
5" and Sage prints "True", then that means that temp really is exactly
the square root of 5.) I use a numeric representation because not all
algebraic numbers can be represented with radicals, and solving by
radicals is much more difficult than the symbolic-numeric algorithms I
used in QQbar.
> So, please tell me. Which should be the right way to try to approach
> this indefinite integration problem? You can see that I'm not that
> good in deep mathematical theory, but approaching the simplest problem
> (that could be different from this one I'm looking at right now) is
> fun :)
Unfortunately, I think the right way is a lot of work.
1) Teach yourself enough math that you can mostly understand the
algorithm in the paper. (Not just mechanically follow the steps, but
understand what each step does, and have some idea why the algorithm
as a whole works.)
For some parts of this task, wikipedia and mathworld.wolfram.com will
be useful resources (planetmath.org is also interersting); but you'll
probably also need to read some books, etc.
2) Implement each missing sub-algorithm in Sage.
3) Implement the entire algorithm in Sage.
This sounds glib, but it really is possible... I'm implementing
cylindrical algebraic decomposition in Sage following this program.
I've been working on it in my spare time for several years; I probably
spent a couple of years on step 1, then I started step 2 a couple of
years ago, and now I'm working on step 2 and step 3.
The problem is that the Risch algorithm just is not simple.
A good integration algorithm probably has a couple of phases -- first
you try some heuristics, and pattern-match against a database of known
integrals; if those fail, then you bring out the big guns and apply
the decision procedure. If you're interested in working on
integration, maybe you'd rather work on the first phase? That
probably requires more computer science and less math than the second
phase.
Carl
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