On Tue, Dec 4, 2018 at 4:23 PM Vincent Delecroix <20100.delecr...@gmail.com> wrote: > > > > Le 04/12/2018 à 15:36, Dima Pasechnik a écrit : > > On Tue, Dec 4, 2018 at 1:47 PM Daniel Krenn <kr...@aon.at> wrote: > >> > >> For integrating a polynomial over a polyhedron LattE is used but if the > >> dimension is not full, then it is not implemented, see > >> > >> sage: x, y = polygens(QQ, 'x, y') > >> sage: P = Polyhedron(vertices=[[0,0],[1,1]]) > >> sage: P.integrate(x*y) # optional - latte_int > >> Traceback (most recent call last): > >> ... > >> NotImplementedError: The polytope must be full-dimensional. > >> > >> from > >> > >> http://doc.sagemath.org/html/en/reference/discrete_geometry/sage/geometry/polyhedron/base.html#sage.geometry.polyhedron.base.Polyhedron_base.integrate > >> > >> I wonder if there is a (simple) way to come around this? > >> > >> [I might not be that familiar with integration over polyhedra, but > >> shouldn't that basically be a somehow "nice" transformation where some > >> Jacobi-determinant comes into play? > > > > you need an orthonormal transformation, thus potentially square roots, > > if you just want to move over to > > P.affine_hull() and integrate there. > > Of course not. There is a well defined *rational* Lebesgue measure on > the affine hull H of your polyhedron. Just consider the Lebesgue measure > so that the lattice Z^d \cap H has covolume 1. With this normalization, > any integral is rational. > Oh, where is the facepalm emoji here...It seems I have to teach multivariate calculus every year in order not to forget how it all works :-[
I suppose that in fact everything needed is actually available in Sage already, just not connected together. Indeed, (A,b)=P.affine_hull(as_affine_map=True, orthogonal=True) gives the rational change of coordinates, and so all you need is to use (A,b) to construct a ring homomorphism from R[x,y] to R[t], apply it to the integrand, integrate the result over P.affine_hull(orthogonal=True) # hopefully it would use the same (A,b) and scale the result by something like the square root of det(AA^T). I'd actually propose to add an option to P.affine_hull() to give the caller both A,b and the resulting polytope. And perhaps there is a 1-line way to construct the ring homomorphism, too (there should be one like this, anyway). Dima > This is what you obtain for the volume (ie integration of the > constant function 1) with the option measure='induced'. > > > But the restriction to full-dimensional isn't really hard to overcome > > with a generalisation of the method > > in LattE, and I see how to do this, I think. > > If anyone cares I could implement this. > > If you do anything please respect the convention naming from the > volume function. > > Vincent > > -- > You received this message because you are subscribed to the Google Groups > "sage-support" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to sage-support+unsubscr...@googlegroups.com. > To post to this group, send email to sage-support@googlegroups.com. > Visit this group at https://groups.google.com/group/sage-support. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at https://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout.
