generic latte_int interface to integrate

[mforets]

Provide a generic interface to integrate a polynomial over a polytope.

See also #18232.

[mforets]

New commits:

​d5ff154	22497: generic interface to LattE integrale count
​96e4099	Integral of a polynomial over a polytope.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​d3c9589

Add volume function to generic latte_int interface.

[vdelecroix]

Why your commit d3c9589 is here!? Shouldn't it belong to another ticket?

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​000bf8b

Restructured the script, with an integrate function accepting different valuations.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​5aa6695

Added test for helper function _to_latte_polynomial.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​72d03a1

Minor changes to helper function and integrate.

[mkoeppe]

Replying to vdelecroix:

Why your commit d3c9589 is here!? Shouldn't it belong to another ticket?

The branch is on top of the branch of #22497.

[mkoeppe]

Looks good to me.

Now follow-up tickets with corresponding methods of Polyhedron would be good.

[git]

Branch pushed to git repo; I updated commit sha1 and set ticket back to needs_review. New commits:

​6a474f9

fixed a typo in the docstring

[mkoeppe]

What does

``[1,[2,2,2]]``

do in the doctest?

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​17911f7

Added integrate method to Polyhedron base.py

[mforets]

Replying to mkoeppe:

What does

``[1,[2,2,2]]``

do in the doctest?

It's me struggling to get tests work in the particular case where a string is passed as argument :(

Probably some issue with string conversion. See this:

In the Sage REPL, the usual single quotes work, as in:

sage: P = polytopes.cuboctahedron()
sage: P.integrate('[[3,[2,4,6]],[7,[0, 3, 5]]]')   
629/47775

(ok I'm already using #20887 for P.integrate(..) but you get the point).

However the doctests didn't pass if I used that, but they pass with the double quote in one test. In Python it seems to have a meaning:

sage: type(``x``)
<type 'str'>

But I'd like to understand what went wrong with just single quotes, of course.

[mkoeppe]

Learn about :: in the doc formatting. This will solve your problem. Also build and verify the html doc.

[tscrim]

There are also a number of doc issues with integrate in base.py. The doc should start with

    def integrate(self, polynomial, **kwds):
        r"""
        Return the integral of a polynomial over a polytope.

        INPUT:

        - ``polynomial`` -- a multivariate polynomial or a valid LattE description string
          for a polynomial

        - ``**kwds`` -- additional keyword arguments that are passed to the engine

        OUTPUT:

        The integral of the polynomial over the polytope.

        .. NOTE::

            The polytope triangulation algorithm is used. This function depends on
            LattE (i.e., the ``latte_int`` optional package).

The code in the EXAMPLES:: block also needs to be indented (as per comment:16).

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​91d885e	Fix the docstrings, and enhance integrate method.
​507aea5	Add the new volume engine Polyhedron.

[mforets]

done! and thanks both for the help. from my side this ticket and #20887 are ok.

in the proposed solution, the new volume and integrate methods can also receive polyhedra in RDF, and each vertex is cast to QQ with an iterator over the vertices.

[vdelecroix]

Replying to mforets:

done! and thanks both for the help. from my side this ticket and #20887 are ok.

Nice work!

in the proposed solution, the new volume and integrate methods can also receive polyhedra in RDF, and each vertex is cast to QQ with an iterator over the vertices.

Is the answer of .integrate() will be a RDF? Otherwise I think it is a bad idea.

[vdelecroix]

Replying to git:

Branch pushed to git repo; I updated commit sha1. New commits:

​91d885e	Fix the docstrings, and enhance integrate method.
​507aea5	Add the new volume engine Polyhedron.

I don't understand why these commits are here... this ticket is about generic latte_int interface to integrate. If you change the purpose of the ticket you need to update the description.

[mforets]

Replying to vdelecroix:

Replying to mforets:

done! and thanks both for the help. from my side this ticket and #20887 are ok.

Nice work!

in the proposed solution, the new volume and integrate methods can also receive polyhedra in RDF, and each vertex is cast to QQ with an iterator over the vertices.

Is the answer of .integrate() will be a RDF? Otherwise I think it is a bad idea.

But why? It will be a QQ.

[vdelecroix]

Replying to mforets:

Replying to vdelecroix:

Replying to mforets:

done! and thanks both for the help. from my side this ticket and #20887 are ok.

Nice work!

in the proposed solution, the new volume and integrate methods can also receive polyhedra in RDF, and each vertex is cast to QQ with an iterator over the vertices.

Is the answer of .integrate() will be a RDF? Otherwise I think it is a bad idea.

But why? It will be a QQ.

It does not make any sense. If I have a Polyhedron defined over RDF (= non exact field) its volume can not defined as an element of QQ (= exact field).

[mforets]

Hmm.. Matrices have a change_ring feature that comes very handy for conversion. Shall we add a similar one for Polyhedron?

And following your recommendation, make integrate and volume via latte raise a value error when instantiated with, say, P = 1.4142*polytopes.cube().

With the change ring, we could still integrate integrate a polynomial over P with the slightly more complicated P.change_ring().integrate(x^2*y^2*z^2), instead of the ugly

sage: P_QQ = Polyhedron(vertices = [[QQ(vi) for vi in v] for v in P.vertices_generator()])
sage: P_QQ.integrate(x^2*y^2*z^2)

If there are alternative recommended options, I'd really like to learn; but I can post it as a ask.sagemath one if i'm off-topic :)

[mkoeppe]

Integration over an RDF polyhedron via QQ polyhedron (latte) could make sense as long as the method converts the answer back to RDF to indicate that the answer is inexact.

As to change_ring, this could be useful. I had related discussions with JP.

[mkoeppe]

(In addition to change_ring, there could also be change_backend.)

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​a563192

fix a bug in _volume_latte