Well, the current FRP systems don't accurately solve this, since they just use an Euler integrator, as do many games. As long as the time steps are tiny enough this usually works good enough. But I wouldn't use these FRPs to guide an expensive robot or spaceship at high precision :-)<div>
<br><div><br><br><div class="gmail_quote">On Tue, Apr 21, 2009 at 11:48 AM, jean-christophe mincke <span dir="ltr"><<a href="mailto:jeanchristophe.mincke@gmail.com">jeanchristophe.mincke@gmail.com</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">Paul,<br><br>Thank you for your reply.<br><br>Integration is a tool to solve a some ODEs but ot all of them. Suppose
all we have is a paper and a pencil and we need to symbolically solve:<br><br><br> / t<br>de(t)/dt = f(t) -> the solution is given by e(t) = | f(t) dt + e(t0)<br>
/ t0<br><br>de(t)/dt
= f(e(t), t) -> A simple integral cannot solve it, we need to use
the dedicated technique appropriate to this type of ODE. <br><br><br>Thus, if the intention of the expression <br><br> e = integrate <i>something </i><br><br>is "I absolutely want to integrate <i>something</i> using some integration scheme", I am not convinced that this solution properly covers the second case above.<br>
<br>However if its the meaning is "I want to solve the ODE : de(t)/dt =<i>something</i> " I would be pleased if the system should be clever enough to analyse the <i>something expression</i> and to apply or propose the most appropriate numerical method.<br>
<br>Since
the two kinds of ODEs require 2 specific methematical solutions, I do
not find suprising that this fact is also reflected in a program.<br>
<br>I have not the same experience as some poster/authors but I
am curious about the way the current FRPs are able to accurately solve
the most simple ODE:<br>
<br>
de(t)/dt = e<br>
<br>
All I have seen/read seems to use the Euler method. I am really interested in knowing whether anybody has implemented a higher order method?<br>
<br>
Regards<br><font color="#888888">
<br>
J-C</font><div><div></div><div class="h5"><br><br><div class="gmail_quote">On Tue, Apr 21, 2009 at 5:03 AM, Paul L <span dir="ltr"><<a href="mailto:ninegua@gmail.com" target="_blank">ninegua@gmail.com</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="border-left:1px solid rgb(204, 204, 204);margin:0pt 0pt 0pt 0.8ex;padding-left:1ex">
Trying to give different semantics to the same declarative definition based<br>
on whether it's recursively defined or not seems rather hack-ish, although<br>
I can understand what you are coming from from an implementation angle.<br>
<br>
Mathematically an integral operator has only one semantics regardless<br>
of what's put in front of it or inside. If our implementation can't match this<br>
simplicity, then we got a problem!<br>
<br>
The arrow FRP gets rid of the leak problem and maintains a single definition<br>
of integral by using a restricted form of recursion - the loop operator.<br>
If you'd rather prefer having signals as first class objects, similar technique<br>
existed in synchronous languages [1], i.e., by using a special rec primitive.<br>
<br>
Disclaimer: I was the co-author of the leak paper [2].<br>
<br>
[1] A co-iterative characterization of synchronous stream functions, P<br>
Caspi, M Pouzet.<br>
[2] Plugging a space leak with an arrow, H. Liu, P. Hudak<br>
<br>
--<br>
Regards,<br>
Paul Liu<br>
<br>
Yale Haskell Group<br>
<a href="http://www.haskell.org/yale" target="_blank">http://www.haskell.org/yale</a><br>
<div><div></div><div><br>
On 4/20/09, jean-christophe mincke <<a href="mailto:jeanchristophe.mincke@gmail.com" target="_blank">jeanchristophe.mincke@gmail.com</a>> wrote:<br>
> In a post in the *Elerea, another FRP library *thread*,* Peter Verswyvelen<br>
> wrote:<br>
><br>
> *>I think it would be nice if we could make a "reactive benchmark" or<br>
> something: some tiny examples that capture the essence of reactive systems,<br>
> and a way to compare each solution's >pros and cons.* *<br>
> *<br>
> *>For example the "plugging a space leak with an arrow" papers reduces the<br>
> recursive signal problem to<br>
> *<br>
> *<br>
> *<br>
> *>e = integral 1 e*<br>
> *<br>
> *<br>
> *>Maybe the Nlift problem is a good example for dynamic collections, but I<br>
> guess we'll need more examples.*<br>
> *<br>
> *<br>
> *>The reason why I'm talking about examples and not semantics is because the<br>
> latter seems to be pretty hard to get right for FRP?*<br>
><br>
> I would like to come back to this exemple. I am trying to write a small FRP<br>
> in F# (which is a strict language, a clone of Ocaml) and I also came across<br>
> space and/or time leak. But maybe not for the same reasons...<br>
><br>
> Thinking about these problems and after some trials and errors, I came to<br>
> the following conclusions:<br>
><br>
> I believe that writing the expression<br>
><br>
> e = integral 1 *something*<br>
><br>
> where e is a Behavior (thus depends on a continuous time).<br>
><br>
> has really two different meanings.<br>
><br>
> 1. if *something *is independent of e, what the above expression means is<br>
> the classical integration of a time dependent function between t0 and t1.<br>
> Several numerical methods are available to compute this integral and, as far<br>
> as I know, they need to compute *something *at t0, t1 and, possibly, at<br>
> intermediate times. In this case, *something *can be a Behavior.<br>
><br>
> 2. If *something *depends directly or indirectly of e then we are faced with<br>
> a first order differential equation of the form:<br>
><br>
> de/dt = *something*(e,t)<br>
><br>
> where de/dt is the time derivative of e and *something*(e,t) indicates<br>
> that *something* depends, without loss of generality, on both e and t.<br>
><br>
> There exist specific methods to numerically solve differential equations<br>
> between t0 and t1. Some of them only require the knowledge of e at t0 (the<br>
> Euler method), some others needs to compute *something *from intermediate<br>
> times (in [t0, t1[ ) *and *estimates of e at those intermediary times.<br>
><br>
> 3. *something *depends (only) on one or more events that, in turns, are<br>
> computed from e. This case seems to be the same as the first one where the<br>
> integrand can be decomposed into a before-event integrand and an after-event<br>
> integrand (if any event has been triggered). Both integrands being<br>
> independent from e. But I have not completely investigated this case yet...<br>
><br>
> Coming back to my FRP, which is based on residual behaviors, I use a<br>
> specific solution for each case.<br>
><br>
> Solution to case 1 causes no problem and is similar to what is done in<br>
> classical FRP (Euler method, without recursively defined behaviors). Once<br>
> again as far as I know...<br>
><br>
> The second case has two solutions:<br>
> 1. the 'integrate' function is replaced by a function 'solve' which has the<br>
> following signature<br>
><br>
> solve :: a -> (Behavior a -> Behavior a) -> Behavior a<br>
><br>
> In fact, *something*(e,t) is represented by an integrand function<br>
> from behavior to behavior, this function is called by the<br>
> integration method. The integration method is then free to pass<br>
> estimates of e, as constant behaviors, to the integrand function.<br>
><br>
> The drawbacks of this solution are:<br>
> - To avoid space/time leaks, it cannot be done without side effects<br>
> (to be honest, I have not been able to find a solution without<br>
> assignement). However these side effects are not visible from outside of the<br>
> solve function. ..<br>
> - If behaviors are defined within the integrand function, they are not<br>
> accessible from outside of this integrand function.<br>
><br>
> 2. Introduce constructions that looks like to signal functions.<br>
><br>
> solve :: a -> SF a a -> Behavior a<br>
><br>
> where a SF is able to react to events and may manage an internal state.<br>
> This solution solves the two above problems but make the FRP a bit more<br>
> complex.<br>
><br>
><br>
> Today, I tend to prefer the first solution, but what is important, in my<br>
> opinion, is to recognize the fact that<br>
><br>
> e = integral 1 *something*<br>
><br>
> really addresses two different problems (integration and solving of<br>
> differential equations) and each problem should have their own solution.<br>
><br>
> The consequences are :<br>
><br>
</div></div>> 1. There is no longer any need for my FRP to be able to define a Behavior<br>
<div>> recursively. That is a good news for this is quite tricky in F#.<br>
> Consequently, there is no need to introduce delays.<br>
</div>> 2. Higher order methods for solving of diff. equations can be used (i.e.<br>
<div><div></div><div>> Runge-Kutta). That is also good news for this was one of my main goal in<br>
> doing the exercice of writing a FRP.<br>
><br>
> Regards,<br>
><br>
> J-C<br>
><br>
</div></div></blockquote></div><br>
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