On Wed, Dec 3, 2008 at 10:38 PM, Creighton Hogg
<wchogg@gmail.com> wrote:
So this is a bit of a brainstorming post. I did a little thinking
earlier about different simple ODE solving techniques, and tried to
translate them fairly directly into Reactive. The result isn't pretty
& I'd like to talk about that.
The naive definitions, to me, of the Euler, Euler-Cromer, and Midpoint
methods should look like the following. Each takes in an Event (),
initial positions & velocities, and a Behavior (v -> v -> v) that is
the acceleration as a function of time. We want to be general since
the acceleration may be an explicit function of time as well as as
position & velocity.
euler tau x0 v0 a = liftA2 (,) x v
where x = accumB x0 (fmap (^+^) (liftA2 (^*) (prev vi) deltaTs))
v = accumB v0 (fmap (^+^) (liftA2 (^*) ai deltaTs))
ai = (snapshot_ tau a) <*> (prev xi) <*> (prev vi)
xi = (snapshot_ tau x)
vi = (snapshot_ tau v)
deltaTs = diffE (tau `snapshot_` time)
eulerCromer tau x0 v0 a = liftA2 (,) x v
where x = accumB x0 (fmap (^+^) (liftA2 (^*) vi deltaTs))
v = accumB v0 (fmap (^+^) (liftA2 (^*) ai deltaTs))
ai = (snapshot_ tau a) <*> (prev xi) <*> (prev vi)
xi = snapshot_ tau x
vi = snapshot_ tau v
deltaTs = diffE (tau `snapshot_` time)
midpoint tau x0 v0 a = liftA2 (,) x v
where x = accumB x0 (fmap (^+^) (liftA2 (^*) vavg deltaTs))
v = accumB v0 (fmap (^+^) (liftA2 (^*) ai deltaTs))
ai = (snapshot_ tau a) <*> (prev xi) <*> (prev vi)
xi = snapshot_ tau x
vi = snapshot_ tau v
deltaTs = diffE (tau `snapshot_` time)
vavg = fmap (^/2) $ liftA2 (^+^) vi (prev vi)
prev :: Event a -> Event a
prev = fmap snd . withPrevE
this code is fairly verbose, but that's because I'm trying to be very
explicit about the ordering of the sampling & how the new values at
each time step are created. The problem is that I feel like I'm
wrestling very hard against Reactive rather than working with it. I
was hoping that I could get some feedback on a way to write these very
simple algorithms that is more inline with the spirit of the API.