On Thu, Dec 14, 2023 at 5:37 AM Nils Bruin <nbr...@sfu.ca> wrote:
>
> (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?
Can't Sage just wrap PARI/gp matsnf (and mathnf) for QQ[t]? They work
for this case too, e.g.
sage: R.<x> = QQ[]
sage: M=matrix(R, [[1+x^2, 2-x+x^3],[42-x, x^3]])
sage: M.__pari__().matsnf(0)
[x^5 + x^4 - 41*x^3 - x^2 + 44*x - 84, 1]
sage: M.__pari__().mathnf(0)
[x^5 + x^4 - 41*x^3 - x^2 + 44*x - 84, 1/74088*x^4 + 43/74088*x^3 +
1765/74088*x^2 + 41/74088*x + 883/37044; 0, 1]
For some reason
sage: M.__pari__().matsnf(0).sage()
cannot deal with conversion back to Sage, which is probably a bug:
NameError Traceback (most recent call last)
Cell In[15], line 1
----> 1 M.__pari__().matsnf(Integer(0)).sage()
File cypari2/gen.pyx:1965, in cypari2.gen.Gen.sage()
File /mnt/opt/Sage/sage-dev/src/sage/libs/pari/convert_sage.pyx:41, in
sage.libs.pari.convert_sage.gen_to_sage()
39
40
---> 41 cpdef gen_to_sage(Gen z, locals=None) noexcept:
42 """
43 Convert a PARI gen to a Sage/Python object.
File /mnt/opt/Sage/sage-dev/src/sage/libs/pari/convert_sage.pyx:300,
in sage.libs.pari.convert_sage.gen_to_sage()
298 return K([gen_to_sage(real), gen_to_sage(imag)])
299 elif t == t_VEC or t == t_COL:
--> 300 return [gen_to_sage(x, locals) for x in z.python_list()]
301 elif t == t_VECSMALL:
302 return z.python_list_small()
File /mnt/opt/Sage/sage-dev/src/sage/libs/pari/convert_sage.pyx:326,
in sage.libs.pari.convert_sage.gen_to_sage()
324 from sage.misc.sage_eval import sage_eval
325 locals = {} if locals is None else locals
--> 326 return sage_eval(str(z), locals=locals)
327
328
File /mnt/opt/Sage/sage-dev/src/sage/misc/sage_eval.py:199, in
sage_eval(source, locals, cmds, preparse)
197 return locals['_sage_eval_returnval_']
198 else:
--> 199 return eval(source, sage.all.__dict__, locals)
File <string>:1
NameError: name 'x' is not defined
>
>
> 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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