Sure, any field is a PID but it is not the targe of a Smith form :)
This question came from an actual computation of a colleague where the
direct approach is taking days and hermite+smith is very fast, and could be
useful for any one variable ring. Any way, I agree with you that maybe a
Summer of Code is a good idea. Until this, a small PR could be helpful?
El jueves, 14 de diciembre de 2023 a las 6:37:29 UTC+1, Nils Bruin escribió:
> (side note: Q is generally considered a PID, but indeed it makes little
> sense worrying about Smith normal form over it)
>
> Concerning the slow performance: from the documentation,
> R._matrix_smith_form would allow an optimized routine to kick in. That
> doesn't exist, though.
> ZZ doesn't have it either, but there is a whole different class for that
> in sage/matrix/matrix_integer_dense.pyx that provides a specific-for-ZZ
> smith form algorithm. For QQ['t'] you end up with the generic algorithm,
> which is liable to be inefficient and have quirks such having large
> variations in runtime, depending on the specific input.
>
> So: it makes sense it's not very optimal (yet?). It would require work to
> get a better implementation specific for QQ['t']. This could be worthwhile
> to do well. Summer of Code?
>
>
> On Thursday 14 December 2023 at 04:05:47 UTC+13 Enrique Artal wrote:
>
>> It is over the ring of polynomials in one variable over Q.
>>
>> El miércoles, 13 de diciembre de 2023 a las 15:05:00 UTC+1, Dima
>> Pasechnik escribió:
>>
>>> On Wed, Dec 13, 2023 at 10:55 AM Enrique Artal <enriqu...@gmail.com>
>>> wrote:
>>> >
>>> > I have the toy example belowto compare the direct use of smith_form
>>> and the combination of hermite_form and smith_form.
>>> >
>>> > Last execution (there are random inputs) gave:
>>> > CPU times: user 2.22 s, sys: 5.93 ms, total: 2.23 s Wall time: 2.23 s
>>> CPU times: user 454 ms, sys: 990 µs, total: 455 ms Wall time: 456 ms
>>> >
>>> > %%%%%%%%%%%%%%%%%%%%%
>>> > R.<t> = QQ[]
>>> > n = 3
>>> > m = 5
>>> > ds = (n, m)
>>> > M = MatrixSpace(R, n, m)
>>> > V = random_vector(R, n)
>>> > A0 = Matrix(R, n, m)
>>> > A0[0, 0] = R.random_element()
>>> > for j in range(1, n):
>>> > A0[j, j] = A0[j -1, j - 1] * R.random_element()
>>> > def par(n):
>>> > i1 = ZZ.random_element(0,n)
>>> > j1 = i1
>>> > while j1 == i1:
>>> > j1 = ZZ.random_element(0,n)
>>> > return (i1, j1)
>>> > def smith_hermite(A):
>>> > D1, U1 = A.hermite_form(transformation=True)
>>> > D, U2, V = D1.smith_form()
>>> > U = U2 * U1
>>> > return D, U, V
>>> > P = {j: identity_matrix(R, ds[j]) for j in range(2)}
>>> > s = 20
>>> > for k in range(2):
>>> > for a in range(s):
>>> > i, j = par(n)
>>> > f = R.random_element()
>>> > P[k].add_multiple_of_row(i, j ,f)
>>> > i, j = par(n)
>>> > P[k].swap_rows(i, j)
>>> > i, j = par(n)
>>> > f = R.random_element()
>>> > P[k].add_multiple_of_column(i, j ,f)
>>> > i, j = par(n)
>>> > P[0].swap_columns(i, j)
>>> > A = P[0] * A0 * P[1]
>>> > %time D, U, V = A.smith_form()
>>> > %time D1, U1, V1 = smith_hermite(A)
>>>
>>> hmm, what's Smith form of a matrix over Q ?
>>> Q is not a PID.
>>> >
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>>>
>>>
>>
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