The speed could be do to the inefficiency of fraction field arithmetic
over the polynomial ring. Ideally, we should have fraction-free
gaussian elimination. Also, easily invertable/small determinant may
actually be worse--as it could be creating a lot of large intermediate
values with non-trivial gcds that are all canceling out.
It would be interesting to profile and see what exactly is taking all
the time.
- Robert
On Dec 17, 2009, at 11:39 AM, Christian Szegedy wrote:
> You evaluate it over ZZ[x1,...,xn] rather than GF(2)[x1,...,x4].
>
> Anyways, it simply can't be *that* slow in any case: even: the
> (theoretically ) maximum number of monoms that can be in any
> expansion is less than a few thousands, so the upper limit
> for a naively implemented Gaussian elimination is in the
> order of seconds. However the matrix is even much simpler than
> that and even the inverse is computed immediately.
>
> On 12/17/09, ma...@mendelu.cz <ma...@mendelu.cz> wrote:
>> And another observation:
>>
>> maxima returns answer immediatelly (with a lag necessary to start
>> maxima)
>> m is the original matrix from x.py
>>
>> sage: m._maxima_().determinant().expand().sage()
>> x0^2*x2^2*x3^2*x7^2 - 2*x0*x1*x2*x3*x4*x5*x6*x7 + x1^2*x4^2*x5^2*x6^2
>>
>>
>> Anyway, the answer is different from expected one. I do not konw
>> which
>> one is correct.
>>
>> Robert
>>
>>
>>
>> On 17 pro, 11:40, "ma...@mendelu.cz" <ma...@mendelu.cz> wrote:
>>> perhaps problems expanding polynomials? even determinant of
>>> submatrix
>>> (0,0,5,5) is suprisingly slow.
>>>
>>> workaroud is to replace polynomials in your matrix by variables.
>>>
>>> var('x0 x1 x2 x3 x4 x5 x6 x7 a1 a2 a3 a4 a5 a6 a7 b1 b2 b3 b4 b5 b6
>>> b7')
>>> m=matrix([[ 0, a1, a2, a3, a4, a5, a6, a7],
>>> [b1, 0, x0, 0, 0, 0, 0, 0],
>>> [b2, x0, 0, x6, 0, 0, 0, 0],
>>> [b3, 0, x6, 0, x3, 0, 0, 0],
>>> [b4, 0, 0, x3, 0, x4, 0, 0],
>>> [b5, 0, 0, 0, x4, 0, x2, 0],
>>> [b6, 0, 0, 0, 0, x2, 0, x1],
>>> [b7, 0, 0, 0, 0, 0, x1,
>>> 0]])
>>> m
>>>
>>> m.det() gives answer immediatelly and you can substitute back for
>>> a's
>>> and b's.
>>>
>>
>>
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