(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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