#18758: Allow actions of a parent on itself
-------------------------------------+------------------------
Reporter: SimonKing | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.8
Component: coercion | Resolution:
Keywords: actions on itself | Merged in:
Authors: | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: #18756 | Stopgaps:
-------------------------------------+------------------------
Comment (by SimonKing):
I haven't pushed my branch yet. Anyway, what I can do is the following:
{{{
sage: class MyElement(Element):
....: def __init__(self, P, L):
....: Element.__init__(self, P)
....: self.l = L
....: def _repr_(self):
....: return "<{}>".format(self.l)
....: def _add_(self, other):
....: return self.parent()(zip(self.l, other.l))
....: def _mul_(self, other):
....: return self.parent()(self.l+other.l)
sage: class MyMElement(ModuleElement):
....: def __init__(self, P, L):
....: ModuleElement.__init__(self, P)
....: self.l = L
....: def _repr_(self):
....: return "<{}>".format(self.l)
....: def _add_(self, other):
....: return self.parent()(zip(self.l, other.l))
....: def _mul_(self, other):
....: return self.parent()(self.l+other.l)
sage: class MyRElement(RingElement):
....: def __init__(self, P, L):
....: RingElement.__init__(self, P)
....: self.l = L
....: def _repr_(self):
....: return "<{}>".format(self.l)
....: def _add_(self, other):
....: return self.parent()(zip(self.l, other.l))
....: def _mul_(self, other):
....: return self.parent()(self.l+other.l)
sage: class MyParent(Parent):
....: Element = MyElement
sage: class MyMParent(Parent):
....: Element = MyMElement
sage: class MyRParent(Parent):
....: Element = MyRElement
sage: P = MyParent(category=Rings())
sage: l = P([1,2])
sage: l*l
<[1, 2, 1, 2]>
sage: PM = MyMParent(category=Rings())
sage: lM = PM([1,2])
sage: lM*lM
<[1, 2, 1, 2]>
sage: PR = MyRParent(category=Rings())
sage: lR = PR([1,2])
sage: lR*lR
<[1, 2, 1, 2]>
sage: %timeit l*l
The slowest run took 4.68 times longer than the fastest. This could mean
that an intermediate result is being cached
100000 loops, best of 3: 8.77 µs per loop
sage: %timeit lM*lM
The slowest run took 6.35 times longer than the fastest. This could mean
that an intermediate result is being cached
100000 loops, best of 3: 7.89 µs per loop
sage: %timeit lR*lR
The slowest run took 7.06 times longer than the fastest. This could mean
that an intermediate result is being cached
100000 loops, best of 3: 7.22 µs per loop
}}}
Explanation:
1. In P, we use Element as starting point. It has no `__mul__`, hence,
`__mul__` is obtained from some element method of the category of rings.
2. In PM, we use !ModuleElement as starting point.
- It has a `__mul__` method that in vanilla Sage would refuse to carry
a ring structure. Hence, the above wouldn't work in vanilla Sage. I have
changed the coercion framework so that it does not complain about a ring
structure.
- Since it has a `__mul__` method, it cannot inherit a ring
multiplication from the category element methods. Instead, the parent
inherits `_get_action_` from parent methods, as per #18756.
3. This would be the "classical" way to define a ring structure.
As you can see, of course a specialised cythoned `__mul__` method for
rings is fastest. The first approach is taken in
`CombinatorialFreeModule`, if I understand correctly, and is slowest (not
by much, but it is). The new approach to use an action of self on itself
somehow is in the middle, speed-wise.
Before I push my branch, I'll see what happens if I provide `Element` with
a default `__mul__` method that simply calls the coercion model. It will
of course make the first approach impossible, but I guess it would speed-
up the second (i.e., new) approach.
--
Ticket URL: <http://trac.sagemath.org/ticket/18758#comment:6>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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