Aha! Quite the challenge is it not?
On Sun, Apr 19, 2009 at 7:44 AM, Maurizio <maurizio.gran...@gmail.com>wrote:
>
> Hi all
>
> > Well, we just need a resultant algorithm that doesn't go through
> > Singular. I'm planning to write such a thing as part of my
> > cylindrical algebraic decomposition implementation sometime in the
> > next few months.
> >
> > Carl
>
> yes, I agree with that.
>
> William, unfortunately I can't understand what the function "variety"
> will help for. Basically, I just wanted to see if I could implement
> the most basic algorithm for rational functions integration, and I
> followed the link in the Pynac wiki. Probably, my mistake was to
> introduce polynomials in this path, but that's the only place where I
> found the "resultant()" function. Do you have any alternative
> suggestion?
>
> 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,y,z> = PolynomialRing(QQ)
> # P.<x,y,z> = GF(5)[]
> P.<x,z> = QQ[]
>
> A = 1
> B = x^3 + x
>
> tores = A - z*diff(B,x)
> res = tores.resultant(B,x); factor(res)
> res1 = res.univariate_polynomial()
> sol1 = res1.roots(ring = QQbar)
>
> with this code, I get the roots over QQbar, which is useful. Then I'd
> like to move to the symbolic field and I do this:
>
> var('x, zs', ns = 1)
> from sage.symbolic.ring import NSR
> As = NSR(A)
> 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?
>
> Assuming to go on (manually for the moment), I do:
> c1 = QQbar(sol1[0][0])
> v1a = A - c1*(3*x^2 + 1)
> Bs.gcd(v1a)
>
> Traceback (click to the left for traceback)
> ...
> RuntimeError: gcd: arguments must be polynomials over the rationals
>
> Traceback (most recent call last):
> File "<stdin>", line 1, in <module>
> File "/home/notebook/sage_notebook/worksheets/admin/4/code/135.py",
> line 9, in <module>
> Bs.gcd(v1a)
> File "/usr/local/sage/local/lib/python2.5/site-packages/
> zope.interface-3.3.0-py2.5-linux-i686.egg/", line 1, in <module>
>
> File "expression.pyx", line 1624, in
> sage.symbolic.expression.Expression.gcd (sage/symbolic/expression.cpp:
> 8608)
> RuntimeError: gcd: arguments must be polynomials over the rationals
>
> My point is: even though I could get the roots in QQbar (which are
> exact), it seems that Pynac is not happy to work with QQbar
> quantities, the only supported seems to be QQ pure rationals.
>
> 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?
>
> 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 :)
>
> Regards
>
> Maurizio
> >
>
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