Re: How to integrate a function over a polygon
- To: mathgroup at smc.vnet.net
- Subject: [mg123326] Re: How to integrate a function over a polygon
- From: Mikael <mikaen.anderson.1969 at gmail.com>
- Date: Thu, 1 Dec 2011 05:54:01 -0500 (EST)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
OK, thanks again for taking the time to reply.
> The result of the division of polynomials of more
> than one variable
> depends on the order of the variables. When you
> specify the order of the
> variables PolynomialReduce uses your order. When you
> don't specify it or
> specify it, it uses the canonical order. For example:
>
> PolynomialReduce[x^2 y + x y^2 + y^2, y + x - 1, {x,
> , y}]
>
> {{y + x y}, y}
>
> PolynomialReduce[x^2 y + x y^2 + y^2, y + x - 1, {y,
> x}]
>
> {{1 + y + x y}, 1 - x}
>
> PolynomialReduce[x^2 y + x y^2 + y^2, y + x - 1]
>
> {{y + x y}, y}
>
> PolynomialReduce[x^2 y + x y^2 + y^2, y + x - 1, x]
>
> {{y + x y}, y}
>
> PolynomialReduce[x^2 y + x y^2 + y^2, y + x - 1, {x,
> x, x}]
>
> {{y + x y}, y}
>
> etc.
>
> Andrzej Kozlowski
>
>
> On 29 Nov 2011, at 13:03, Mikael wrote:
>
> > Thanks Andrzej but that is exactly the part in
> PolynomialReduce which
> I do not understand. As I mentioned you get also
> exactly the same answer
> if you change the last argument to {z, x} or {z}. So
> I would appreciate
> if someone could explain the meaning of the last
> argument in
> PolynomialReduce.
> >
> > In a related question I wonder how one can plot
> g[x,y] over only the
> 2-dimensional unit simplex.
> >
> > Many thanks to all in advance.
> >
> > /Mikael
> >
> >> Actually, that was a mistake ;-) It should have
> been
> >>
> >> f[x_, y_, z_] := x^2 + y^2 + z^2
> >>
> >> Expand[
> >> Last[PolynomialReduce[f[x, y, z], {x + y + z - 1},
> >> , {z, x, y}]]]
> >>
> >> 1 - 2 x + 2 x^2 - 2 y + 2 x y + 2 y^2
> >>
> >> But luckily it made not difference ;-)
> >>
> >>
> >> In this particular case PolynomialReduce was an
> >> overkill. You could
> >> equally well use a simple replacement:
> >>
> >> g[x_,y]:=x^2 + y^2 + z^2 /. z -> (1 - x - y) //
> >> Expand
> >>
> >> 1 - 2 x + 2 x^2 - 2 y + 2 x y + 2 y^2
> >>
> >>
> >> PolynomialReduce could be useful if you wanted to
> >> integrate over a more
> >> complicated non-linear regions, when a simple
> >> syntactical substitution
> >> might not be possible.
> >>
> >> Andrzej Kozlowski
> >>
> >>
> >> On 27 Nov 2011, at 10:14, Mikael wrote:
> >>
> >>> Many thanks indeed for your elegant solution,
> >> Andrzej. May I ask a
> >> related question regarding the last argument in
> >>>
> >>> Expand[Last[PolynomialReduce[f[x, y, z], {x + y +
> z
> >> - 1}, {z, x,
> >> z}]]].
> >>>
> >>> I wonder what is the role of {z, x, z} there. I
> get
> >> the same answer if
> >> I change it to {z, x} or {z} and I could not
> figure
> >> it out from the help
> >> page for PolynomialReduce either.
> >>>
> >>> /Mikael
> >>>
> >>>
> >>>
> >>>
> >>>> Well, perhaps you mean this. Let the function
> be:
> >>>>
> >>>> f[x_, y_, z_] := x^2 + y^2 + z^2
> >>>>
> >>>> We want to integrate it over the simplex:
> >> x+y+z==1,
> >>>> 0<=x<=1,0<=y<=1,0<=y<=1
> >>>>
> >>>> On the simplex the function can be expressed in
> >> terms
> >>>> of only x and y as follows:
> >>>>
> >>>> g[x_, y_] =
> >>>> Expand[Last[PolynomialReduce[f[x, y, z], {x + y
> +
> >> z
> >>>> z - 1}, {z, x, z}]]]
> >>>>
> >>>> 2*x^2 + 2*x*y - 2*x + 2*y^2 - 2*y + 1
> >>>>
> >>>> In terms of x and y the simplex can be described
> >> as:
> >>>>
> >>>> cond[x_, y_] := x + y <= 1 && 0 <= x <= 1 && 0
> <=
> >> y
> >>>> <= 1
> >>>>
> >>>> So now we simply compute:
> >>>>
> >>>> Integrate[Boole[cond[x, y]]*g[x, y], {x, 0, 1},
> >> {y,
> >>>> 0, 1}]
> >>>>
> >>>> 1/4
> >>>>
> >>>>
> >>>> Andrzej Kozlowski
> >>>>
> >>>>
> >>>> On 25 Nov 2011, at 10:57, Mikael wrote:
> >>>>
> >>>>> Well, as I wrote in my OP, it is a
> 2-diemnsional
> >>>> unit simplex so you can always re-parametrize
> the
> >>>> function to have 2 arguments.
> >>>>>
> >>>>> In any case, your answer is not useful unless
> you
> >>>> had also answered the original question apart
> from
> >>>> your remark.
> >>>>>
> >>>>>> First of all, f would need three arguments.
> >>>>>>
> >>>>>> Bobby
> >>>>>>
> >>>>>> On Wed, 23 Nov 2011 06:07:00 -0600, Mikael
> >>>>>> <mikaen.anderson.1969 at gmail.com> wrote:
> >>>>>>
> >>>>>>> The subject line asks the general question
> but
> >> to
> >>>>>> be more specific
> >>>>>>> suppose I have a 2-dimentional unit simplex
> >>>> defined
> >>>>>> as
> >>>>>>>
> >>>>>>> Polygon[{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}].
> >>>>>>>
> >>>>>>> I winder how I can integrate a function
> f(x,y)
> >>>> over
> >>>>>> this simplex. Thanks.
> >>>>>>>
> >>>>>>
> >>>>>>
> >>>>>> --
> >>>>>> DrMajorBob at yahoo.com
> >>>>>>
> >>>>>
> >>>>
> >>>>
> >>>
> >>
> >>
> >
>
>