Martin Albrecht wrote:
> On Thursday 02 April 2009, Christophe Oosterlynck wrote:
>> Hi,
>>
>> is it possible to calculate the echelon form of matrices containing
>> polynomials over GF(2)? I'm getting the following error:
>>
>> NotImplementedError: echelon form over Univariate Polynomial Ring in x
>> over Finite Field of size 2 (using NTL) not yet implemented
>>
>> M._maxima_().echelon() does calculate an echelon form but it takes a
>> while and it's not taking into account the finite field GF(2)...
>
> Hi,
>
> one question of course is how you define the echelon form over a ring. If you
> construct a *multi*variate polynomial ring over GF(2) in *one* variable, then
> you have three options:
>
> sage: P = PolynomialRing(GF(2),1,'x')
> sage: type(P)
> <type
> 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular'>
> sage: M = random_matrix(P,5,5)
> sage: M
>
> [ x^2 + 1 x^2 x^2 + x x^2 + x + 1 x + 1]
> [ 0 x^2 + x x 0 x^2]
> [ x + 1 x^2 + x + 1 1 0 x + 1]
> [ x^2 x^2 + x + 1 x^2 + x + 1 0 x + 1]
> [ x^2 + x 1 x^2 + 1 x^2 + x x^2 + 1]
> sage: type(M)
> <type 'sage.matrix.matrix_mpolynomial_dense.Matrix_mpolynomial_dense'>
>
> sage: M.echelon_form('row_reduction')
> sage: M.echelon_form('frac')
> sage: M.echelon_form('bareiss')
>
> Check
>
> sage: M.echelon_form?
>
> to see what each does in detail, basically:
>
> * row_reduction: only reduce by base field elements
> * frac: compute in the fraction field
> * bareiss: fraction free Gauss elimination (the name is rather bad)
This might also help resolve the back-burner project of mine to make
echelon_form for integer matrices return the typical linear algebra
definition of rref, i.e., the 'frac' method above. Currently, the
decision is to make echelon_form behave like 'frac' above, and make
hermite_form behave like 'row_reduction' above. Making echelon_form
support the arguments above seems like it would be a better resolution
and provide a clear deprecation path. However, the decision would imply
that the default be 'frac' instead of 'row_reduction'. On the other
hand, it'd be great if things were consistent across base rings. What
is the possibility of making the default to be 'frac' over polynomial
rings? (Sorry if this question is laughable or repulsive...)
Jason
--~--~---------~--~----~------------~-------~--~----~
To post to this group, send email to [email protected]
To unsubscribe from this group, send email to
[email protected]
For more options, visit this group at
http://groups.google.com/group/sage-support
URLs: http://www.sagemath.org
-~----------~----~----~----~------~----~------~--~---
