On Tue, Jun 18, 2013 at 11:08 PM, Simon King <[email protected]> wrote:
> Hi Fredrik,
>
> On 2013-06-18, Fredrik Johansson <[email protected]> wrote:
>> Hi all,
>>
>> Version 0.1 of the ore_algebra package for Sage has been released:
>> http://www.risc.jku.at/research/combinat/software/ore_algebra/
>>
>> The package is being developed by Manuel Kauers, Maximilian Jaroschek
>> and myself. A tutorial paper (as well as Sphinx documentation) is
>> available on the web site.
>
> Out of curiosity: Does it use the non-commutative stuff (g-algebras)
> that is in Singular and that is also wrapped in Sage?
>
> For the latter, see the documentation of
> sage.rings.polynomial.plural.g_Algebra
> (that's a factory) respectively
> sage.rings.polynomial.plural.NCPolynomialRing_plural
> (that's the wrapper)?
>
> If I'm not mistaken, g-algebras are general enough to implement Ore
> algebras. If your code is not based on Singular: How does it compare,
> speed-wise?
>
> Best regards,
> Simon
No, we just use commutative Sage polynomials.
Here is a test of arithmetic with ore_algebra in Sage 5.9:
sage: A.<x> = QQ[]
sage: R.<Dx> = OreAlgebra(A, 'Dx')
sage: a = (Dx + x + 1)
sage: %timeit a * a
1000 loops, best of 3: 280 us per loop
sage: a = (Dx + x + 1)^10
sage: %timeit a * a
10 loops, best of 3: 28.9 ms per loop
sage: a = (Dx + x + 1)^100
sage: %timeit a * a
1 loops, best of 3: 11.2 s per loop
With g_algebra:
sage: A.<x,Dx> = FreeAlgebra(QQ,2)
sage: R.<x,Dx> = A.g_algebra(relations={Dx*x:x*Dx+1}, order='lex')
sage: a = (Dx + x + 1)
sage: %timeit a * a
1000000 loops, best of 3: 1.69 us per loop
sage: a = (Dx + x + 1)^10
sage: %timeit a * a
100 loops, best of 3: 7.05 ms per loop
sage: a = (Dx + x + 1)^100
sage: %timeit a * a
1 loops, best of 3: 715 s per loop
So it appears that the Python overhead makes our code quite a bit
slower for small objects, while we scale better for large objects, or
maybe the multiplication in Plural just does something stupid...
Fredrik
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