#4681: [with patch, needs work] General Smith normal form implementation
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Reporter: davidloeffler | Owner: davidloeffler
Type: enhancement | Status: assigned
Priority: major | Milestone: sage-3.2.2
Component: linear algebra | Resolution:
Keywords: matrix, smith normal form |
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Changes (by was):
* summary: [with patch, needs review] General Smith normal form
implementation => [with patch, needs work]
General Smith normal form implementation
Comment:
I read the code, which looks fine. The doctests pass. This test passes:
{{{
for i in range(10):
print i
a = random_matrix(ZZ[sqrt(-1)],7)
d,u,v = a.smith_form()
assert u*a*v == d
assert u.det() == 1
assert v.det() == 1
}}}
This fails pretty quickly, but it's not the fault of this code:
{{{
K.<a>=NumberField(x^3-2)
print "h = ", K.class_number()
for i in range(10):
print i
a = random_matrix(K,3)
d,u,v = a.smith_form()
assert u*a*v == d
assert u.det() == 1
assert v.det() == 1
...
Traceback (most recent call last): d,u,v = a.smith_form()
File "element.pyx", line 1269, in
sage.structure.element.CommutativeRingElement.inverse_mod
(sage/structure/element.c:10039)
NotImplementedError
}}}
It's the fault of other stuff missing in Sage.
Another thing -- in matrices over ZZ there is a smith_form function, and
it's inputs, etc., are *not* consistent with the above. It doesn't have a
transformation option - instead it always gives the transformation
matrices. However, it also says to use the function "elementary_divisors"
to get the diagonal entries. I attached a patch that fixes this by
making the sage integer matrix snf have the same interface as the generic
one.
The definition of smith_form between your new code and Sage's code (which
relies on pari) is inconsistent (see below). This needs to be somehow
resolved. I haven't done this yet at all.
{{{
sage: a = matrix(ZZ,3,[1..9]); a
[1 2 3]
[4 5 6]
[7 8 9]
sage: a.smith_form()
([0 0 0]
[0 3 0]
[0 0 1],
[ 1 -2 1]
[ 0 -1 1]
[ 0 2 -1],
[ 1 2 -1]
[-2 -1 1]
[ 1 0 0])
sage: a.smith_form(transformation=False)
[0 0 0]
[0 3 0]
[0 0 1]
sage: a = matrix(ZZ[i],3,[1..9]); a
[1 2 3]
[4 5 6]
[7 8 9]
sage: a.smith_form(transformation=False)
[ 1 0 0]
[ 0 -3 0]
[ 0 0 0]
}}}
To get this into Sage, please:
* review my matrix_integer_dense patch
* decide on what to do about the inconsistency between sage/pari/your
code.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/4681#comment:7>
Sage <http://sagemath.org/>
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