Opened 12 years ago

Action of MatrixGroup on a MPolynomialRing

Reported by: Owned by: SimonKing SimonKing major sage-6.4 commutative algebra matrix group, action, polynomial ring wdj, malb Simon King David Loeffler, William Stein N/A

A group of n by n matrices over a field K acts on a polynomial ring with n variables over K. However, this is not implemented yet.

The following should work:

```sage: M = Matrix(GF(3),[[1,2],[1,1]])
sage: N = Matrix(GF(3),[[2,2],[2,1]])
sage: G = MatrixGroup([M,N])
sage: m = G.0
sage: n = G.1
sage: R.<x,y> = GF(3)[]
sage: m*x
x + y
sage: x*m
x - y
sage: (n*m)*x == n*(m*x)
True
sage: x*(n*m) == (x*n)*m
True
```

On the other hand, we still want to have the usual action on vectors or matrices:

```sage: x = vector([1,1])
sage: x*m
(2, 0)
sage: m*x
(0, 2)
sage: (n*m)*x == n*(m*x)
True
sage: x*(n*m) == (x*n)*m
True
```
```sage: x = matrix([[1,2],[1,1]])
sage: x*m
[0 1]
[2 0]
sage: m*x
[0 1]
[2 0]
sage: (n*m)*x == n*(m*x)
True
sage: x*(n*m) == (x*n)*m
True
```

comment:1 Changed 12 years ago by SimonKing

• Component changed from algebra to commutative algebra
• Owner changed from tbd to malb

Sorry, in the above code I forgot to copy/paste the line

```sage: R.<x,y> = GF(3)[]
```

Moreover, for the reasons above, the ticket should belong to *commutative* algebra, not just algebra (I was clicking on the wrong button).

comment:2 Changed 12 years ago by SimonKing

• Summary changed from Action of MatrixGroup on a MPolynomialRing to [with patch, needs work] Action of MatrixGroup on a MPolynomialRing

The patch `matrixgroupCall.patch` is without doctests, and I am not sure if it couldn't be done better. So, it needs more work.

For example, Martin mentioned the possibility (off list) to create a pyx file with a Cython function, and then the call method would use that function. It might pay off here, since there are tight loops and since the method has to deal with tuples or lists. So Cdefining might speed things up.

Opinions?

At least, the following now works:

```sage: R.<x,y>=GF(3)[]
sage: M=Matrix(GF(3),[[1,2],[1,1]])
sage: G=MatrixGroup([M])
sage: g=G.0
sage: p=x*y^2
sage: g(p)
x^3 + x^2*y - x*y^2 - y^3
```

comment:3 Changed 12 years ago by mabshoff

• Milestone set to sage-3.2.1

Changed 12 years ago by SimonKing

call method for MatrixGroupelement? (this time with doc test) and left_matrix_action for MPolynomial

comment:4 Changed 12 years ago by SimonKing

• Owner changed from malb to SimonKing
• Summary changed from [with patch, needs work] Action of MatrixGroup on a MPolynomialRing to [with patch, needs review] Action of MatrixGroup on a MPolynomialRing

Some changes:

Nicolas Thiéry suggested to implement the action of matrix groups by a method for polynomials (I call the method `left_matrix_action`), and, for convenience, also provide a `__call__` method for MatrixGroupElement? that refers to `left_matrix_action`.

• The `__call__` works for any class that has a `left_matrix_action` method, hence, it is not a-priori restricted to polynomials.
• I've put `left_matrix_action` into `multi_polynomial.pyx`, hence, I can use Cython.

I have two Cython concerns left:

1. The innerst loop is
```prod([Im[k]**X[k] for k in xrange(n)])
```
where `k` is c'defined as `int`. Should this better be done in a for-loop, rather then creating a list and calling `prod`?
2. The variable `X` is of type `polydict.ETuple`, so I can not directly c'define `X`. One could do
```   cdef tuple X
for i from 0<i<l:
X = tuple(Expo[i])
```
But would this be faster?

Changed 12 years ago by SimonKing

Another version of left_matrix_action

comment:5 follow-up: ↓ 6 Changed 12 years ago by SimonKing

In `matrixgroupCallNew.patch` (to be applied after the first patch), I modified the method according to my above concerns. In the example from my original post, the average running time improves from ~240 microseconds to 164 microseconds, and in a larger example it improved from 6.5s to 5.4s

Nevertheless, I made two separate patches, so that the reviewer (if there is any...) can compare by him- or herself.

Cheers

Simon

comment:6 in reply to: ↑ 5 ; follow-up: ↓ 7 Changed 12 years ago by SimonKing

One observation: Reverse the outer loop

```        for i from l>i>=0:
X = tuple(Expo[i])
c = Coef[i]
for k from 0<=k<n:
if X[k]:
c *= Im[k]**X[k]
q += c
```

It results in a further improvement of computation time. Is this coincidence? Or is it since summation of polynomials should better start with the smallest summands?

Anyway, I didn't change the patch yet.

Changed 12 years ago by SimonKing

Slight improvement; extended functionality

comment:7 in reply to: ↑ 6 Changed 12 years ago by SimonKing

One observation: Reverse the outer loop

```        for i from l>i>=0:
X = tuple(Expo[i])
c = Coef[i]
for k from 0<=k<n:
if X[k]:
c *= Im[k]**X[k]
q += c
```

It results in a further improvement of computation time. Is this coincidence? Or is it since summation of polynomials should better start with the smallest summands?

I made a couple of tests, and there was a small but consistent improvement. So, in the third patch (to be applied after the other two) I did it in that way.

The `left_matrix_action` shall eventually be used for computing the Reynolds operator of a group action; moreover, the Reynolds operator should be applicable on a list of polynomials. Then, the function would repeatedly compute the image of the ring variables under the action of some group element. But then it would be better to compute that image only once and pass it to `left_matrix_action`. The new patch provides this functionality. Example (continuing the original example):

```sage: L=[X.left_matrix_action(g) for X in R.gens()]
sage: p.left_matrix_action(L)
x^3 + x^2*y - x*y^2 - y^3
```

comment:8 Changed 12 years ago by wdj

I did confirm that the patches apply cleanly, that

```sage: M = Matrix(GF(3),[[1,2],[1,1]])
sage: G = MatrixGroup([M])
sage: g = G.0
sage: g

[1 2]
[1 1]
sage: P.<x,y> = PolynomialRing(GF(3),2)
sage: p = x*y^2
sage: g(p)
x^3 + x^2*y - x*y^2 - y^3
sage: (x+2*y)*(x+y)^2
x^3 + x^2*y - x*y^2 - y^3

```

works, and that the code seems well-documented.

However, I can't do testing on this machine (intrepid ubuntu) and some of the code is written in Cython, which I can't really 100% vouch for. Seems okay though and simple enough. Since speed was a topic of the comments above, my only question is that the segment

``` 	396	        for i from 0<=i<l:
397	            X = Expo[i]
398	            c = Coef[i]
399	            q += c*prod([Im[k]**X[k] for k in xrange(n)])
```

could probably be rewritten as a one-line sum, which might (or might not) be faster.

Maybe Martin Albrecht could comment on the Cython code?

If Martin (for example) passes the Cython code, and the docstrings pass sage -testall, I would give it a positive review.

comment:9 Changed 12 years ago by malb

```cdef list Im
if isinstance(M,list):
Im = M
```

shouldn't Im = M take care of the type checking anyway, so that a try-except block is sufficient? Also, I think maybe the type of p should be checked in the `__call__` method and a friendly error message raised? Not sure though.

comment:10 Changed 12 years ago by malb

Cython code looks good (just read it).

comment:11 Changed 12 years ago by was

• Summary changed from [with patch, needs review] Action of MatrixGroup on a MPolynomialRing to [with patch, needs work] Action of MatrixGroup on a MPolynomialRing

REFEREE REPORT:

Check this out:

```sage: R.<x,y> = GF(3)[]
sage: M=Matrix(GF(3),[[1,2],[1,1]])
sage: M2=Matrix(GF(3),[[1,2],[1,0]])
sage: G=MatrixGroup([M, M2])
sage: (G.0*G.1)(p)
-x^2*y + x*y^2 - y^3
sage: G.0(G.1(p))
x^2*y + x*y^2 + y^3
```

Oops, your *left action* -- which it better be if you use that notation -- ain't a left action! Oops

-- William

comment:12 follow-up: ↓ 13 Changed 12 years ago by SimonKing

Really Oops. Sorry.

I implemented it analogous to what is done in Singular. Perhaps I am mistaken in the sense that it is supposed to be a right action (which then would deserve another notation).

```sage: (G.0(G.1((p))))
-x^2*y + x*y^2 - y^3
sage: (G.1*G.0)(p)
-x^2*y + x*y^2 - y^3
```

However, I think it doesn't matter what Singular does. I will look up the literature whether one really wants a right or left action, and how the left action is supposed to be.

comment:13 in reply to: ↑ 12 Changed 12 years ago by SimonKing

However, I think it doesn't matter what Singular does. I will look up the literature whether one really wants a right or left action...

I mean, something like "one has a left action on a variety, which gives rise to a right action on the coordinate ring". I have to sort it out.

If this is the case, then it should be better implemented in the `__mul__` method of polynomials, isn't it? Such as

```sage: p*G.1*G.0==p*(G.1*G.0)
True
```

comment:14 Changed 12 years ago by was

Left actions should use call, right actions should use *exponentiation*.

Substitution is a right action. Substitution of the *inverse* is a left action.

comment:15 Changed 11 years ago by SimonKing

• Description modified (diff)
• Report Upstream set to N/A

Changed 11 years ago by SimonKing

Replaces the other patches

comment:16 Changed 11 years ago by SimonKing

• Authors set to Simon King
• Status changed from needs_work to needs_review

The new patch solves the problems addressed in the new ticket description.

I worked on top of several other tickets, since I somehow cared about number fields. To be precise, I did

```hg_sage.import_patch('http://trac.sagemath.org/sage_trac/raw-attachment/ticket/9205/trac_9205-discrete_log.patch')
hg_sage.import_patch('http://trac.sagemath.org/sage_trac/raw-attachment/ticket/9205/trac_9205-doctest.patch')
hg_sage.import_patch('http://trac.sagemath.org/sage_trac/raw-attachment/ticket/9438/trac_9438_IntegerMod_log_vs_PARI.patch')
hg_sage.import_patch('http://trac.sagemath.org/sage_trac/raw-attachment/ticket/9423/trac_9423_gap_for_numberfields.patch')
hg_sage.import_patch('http://trac.sagemath.org/sage_trac/raw-attachment/ticket/8909/8909_gap2cyclotomic.patch')
hg_sage.import_patch('http://trac.sagemath.org/sage_trac/raw-attachment/ticket/8909/trac_8909_catch_exception.patch')
hg_sage.import_patch('http://trac.sagemath.org/sage_trac/raw-attachment/ticket/5618/trac_5618_gap_for_cyclotomic_fields.patch')
```

before creating my pacth. But this shouldn't matter, I guess the patch applies cleanly to `sage-4.4`.

comment:17 Changed 11 years ago by SimonKing

• Summary changed from [with patch, needs work] Action of MatrixGroup on a MPolynomialRing to Action of MatrixGroup on a MPolynomialRing

comment:18 Changed 11 years ago by SimonKing

I made Cc to malb and wdj, since it both concerns polynomials and groups.

comment:19 follow-up: ↓ 20 Changed 10 years ago by nharris

• Status changed from needs_review to needs_work

Sorry to be a nag, but matrix_group_element.py doesn't pass the doctests:

```----------------------------------------------------------------------
matrix_group_element.py
SCORE matrix_group_element.py: 87% (14 of 16)

Missing documentation:
* is_MatrixGroupElement(x):
* __invert__(self):

----------------------------------------------------------------------
```

This is the same doctest score that the old unpatched file gets too.

comment:20 in reply to: ↑ 19 Changed 10 years ago by SimonKing

• Status changed from needs_work to needs_review

Sorry to be a nag, but matrix_group_element.py doesn't pass the doctests:

Which one fails?

```----------------------------------------------------------------------
matrix_group_element.py
SCORE matrix_group_element.py: 87% (14 of 16)

Missing documentation:
* is_MatrixGroupElement(x):
* __invert__(self):

----------------------------------------------------------------------
```

This is the same doctest score that the old unpatched file gets too.

Of course there is no change. I added code to one already existing method, extending its functionality, and added tests for the new functionality. But this patch is not about inversion of matrix group elements, and I think the patch is not supposed to add documentation to methods that are not in its scope.

So, unless a doc test fails because of my patch, the criticism about not raising the doc test coverage is invalid.

comment:21 follow-up: ↓ 22 Changed 10 years ago by nharris

I'm sorry if I did something that I shouldn't have. I was just following this guideline for reviewing patches (found here http://www.sagemath.org/doc/developer/trac.html#section-review-patches):

Is it documented sufficiently, including both explanation and doctests? This is very important: all code in Sage must have doctests, so even if the patch is for code which did not have a doctest before, the new version must include one. In particular, all new code must be 100% doctested. Use the command sage -coverage <files> to see the coverage percentage of <files>.

comment:22 in reply to: ↑ 21 Changed 10 years ago by SimonKing

Is it documented sufficiently, including both explanation and doctests? This is very important: all code in Sage must have doctests, so even if the patch is for code which did not have a doctest before, the new version must include one. In particular, all new code must be 100% doctested. Use the command sage -coverage <files> to see the coverage percentage of <files>.

If you would read the patch, you would find:

1. The patch adds one case to an existing method, namely `_act_on_`.
1. The patch adds several doc tests to `_act_on_`, covering the new functionality.
1. The original version of `_act_on_` already had a doc test.

In particular, it is impossible to detect the doc test change without reading the patch, by just using `sage -coverage`: The coverage script detects whether `_act_on_` has any test (this is the case with or without my patch), but it does not detect whether the patch extends existing documentation to cover a new case.

comment:23 Changed 10 years ago by davidloeffler

Apply `trac-4513_matrix_action_on_polynomials.patch`

(for patchbot -- it's trying to apply all the patches at once)

comment:24 Changed 10 years ago by davidloeffler

`Apply trac-4513_matrix_action_on_polynomials.patch`

(maybe it will work this time?)

comment:25 Changed 10 years ago by davidloeffler

```Apply trac-4513_matrix_action_on_polynomials.patch
```

(For some reason patchbot's not picking this up -- I apologise to all human beings reading this for the spam!)

comment:26 Changed 10 years ago by davidloeffler

• Reviewers set to David Loeffler, William Stein
• Status changed from needs_review to needs_work

I've had a look at the patch, and I don't think you've addressed William's comment #14 from two years back. The following makes me *extremely* uneasy:

```sage: G = GL(3, 7)
sage: R.<a, b> = GF(7)[]
sage: G.0 * a
[3*a   0   0]
[  0   a   0]
[  0   0   a]
sage: R.<a,b,c> = GF(7)[]
sage: G.0 * a
3*a
```

It looks like there's some pre-existing coercion mechanism which returns elements of the matrix space over R, and you're overriding it in one case with an alternative coercion that returns completely different answers; this violates a Sage coercion axiom (where there are multiple paths in the coercion diagram, all must give the same answer up to numerical precision issues). Moreover, if you look at the patchbot logs it seems to have found an example where the preexisting coercion gets picked up instead of the new one.

Sorry, that's a thumbs down from me.

David

comment:27 Changed 8 years ago by jdemeyer

• Milestone changed from sage-5.11 to sage-5.12

comment:28 Changed 7 years ago by vbraun_spam

• Milestone changed from sage-6.1 to sage-6.2

comment:29 Changed 7 years ago by vbraun_spam

• Milestone changed from sage-6.2 to sage-6.3

comment:30 Changed 7 years ago by vbraun_spam

• Milestone changed from sage-6.3 to sage-6.4
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