Challenging? Just because about the problem of integration in finite terms Hardy in 1916 stated that “there is reason to suppose that no such method can be given” ? :)
I want to add to this discussion that I found a lot of useful information in this thread from SymPy list: http://groups.google.com/group/sympy/browse_frm/thread/47259e49ad1cfd13/7a98521ffb13e311?lnk=gst&q=risch#7a98521ffb13e311 I'm wondering whether it could be a good idea to add to the Pynac wiki a link to this article from Manuel Bronstein: http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac98.pdf This seems a good reading! Moreover, in the same thread, I read that a lot of work has been accomplished in SymPy. I know I can sometimes sound like a broken record (always repeating the same crap...), but I more and more see like a great advance in taking advantage of SymPy's python code. I am willing to go and read it, and hopefully I can catch at least a single bit from it :P I want to point out that this (in my opinion) is just to give pynac a fastest bootstrap, so that we can start working on improving something already there. Unfortunately, porting some of SymPy's code sounds more like a Computer Science task, which is not really my field. If I look at my first aim (Laplace and Fourier transforms...) I can see a long way to cover, hopefully in the shortest time frame! Regards Maurizio On 19 Apr, 17:10, Martin Musatov <marty.musa...@gmail.com> wrote: > 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 --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
