2009/6/15 Jean-Guillaume Dumas jean-guillaume.du...@imag.fr:
William Stein wrote:
On Wed, Jun 10, 2009 at 6:03 PM, Yannyannlaiglecha...@gmail.com wrote:
--
| Sage Version 4.0.1, Release Date: 2009-06-06 |
| Type notebook() for the GUI, and license() for information. |
--
sage: A=matrix(GF(3),2,[0,0,1,2])
sage: R.x=GF(3)[]
sage: D={ x:0 , x+1:0 , x^2+x:0 }
sage: for i in range(10):
: D[A._minpoly_linbox()]+=1
:
sage: D
{x: 38266, x + 1: 29397, x^2 + x: 32337}
You're absolutely right! This *sucks* -- it seems like nothing we
have ever wrapped in Linbox is right at first. Hopefully the issue is
that somehow the algorithm is only supposed to be probabilistic, and
we're just misusing it in sage (quite possible).
Anyway, Clement Pernet will be at Sage Days next week, and we'll sort this
out.
Many thanks for brining this to our attention!
This is now:
http://trac.sagemath.org/sage_trac/ticket/6296
Well, I think this was corrected in linbox-1.1.6:
We're using linbox-1.1.6 in Sage.
The minpoly algorithm used depends on which method you are using from
LinBox of course but,
If you use the solution minpoly you will get the blackbox algorithm
(just like if you specify minpoly(pol, mat, Method::Blackbox()))
then (since sept 2008 and 1.1.6) we will end up using an extension field
to compute the minpoly (on my machine it will be GF(3^10)) and then
I e.g. got the following result for one try (the algorithm is still
probabilistic, but has a much larger success rate, roughly around 1/3^10):
Here's what we're using:
void linbox_modn_dense_minpoly(mod_int modulus, mod_int **mp, size_t*
degree, size_t n, mod_int **matrix, int do_minpoly) {
ModInt F((double)modulus);
size_t m = n;
DenseMatrixModInt A(linbox_new_modn_matrix( modulus, matrix, m, m));
GivPolynomialModInt::Element m_A;
if (do_minpoly)
minpoly(m_A, A);
else
charpoly(m_A, A);
(*mp) = new mod_int[m_A.size()];
*degree = m_A.size() - 1;
for (size_t i=0; i = *degree; i++) {
(*mp)[i] = (mod_int)m_A[i];
}
}
This is from the file interfaces/linbox-sage.C, which ships with linbox.
Many thanks for clarifying that minpoly fails with some probability,
and that we need to call it multiple times, take lcm's, and force the
user to give the option proof=False to use it.
Just out of curiosity, is there any provably correct minpoly in
linbox? We don't have one in Sage at all, so it would be useful so
we don't have to implement one from scratch.
William
3 minimal Polynomials are x^2 +x, 3 minimal polynomial are x+1, 4
minimal polynomials are x
Now for a so small matrix it could be better to use a dense version,
which can be called by minpoly(pol,mat,Method::Elimination()).
If i am correct this dense version is also probabilistic (choice of the
Krylov non-zero vector) and therefore should also pick vectors from an
extension.
This is not the case in 1.1.6.
Clément can you confirm this ? If so it should be easy to fix, the same
way we fixed Wiedemann.
For your example matrix in some of the cases, when vectors [1,1], and
[2,2] are chosen the Krylov space has rank 1, whereas for other non zero
vectors it has rank 2 and
thus the dense minbpoly will be x^2+x or x+1 ...
btw, the returned polynomial is always a factor of the true polynomial,
therefore to get a 1/3^{10k} probability of success it will be
sufficient to perform the lcm of k runs.
Best,
--
Jean-Guillaume Dumas.
jean-guillaume.du...@imag.fr Tél.: +33 476 514 866
Université Joseph Fourier, Grenoble I. Fax.: +33 476 631 263
Laboratoire Jean Kuntzmann, Mathématiques Appliquées et Informatique
51, avenue des Mathématiques. LJK/IMAG - BP53. 38041 Grenoble FRANCE
http://ljk.imag.fr/membres/Jean-Guillaume.Dumas
--
William Stein
Associate Professor of Mathematics
University of Washington
http://wstein.org
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