From: Todd Benson <caduceass@...>
Date: 2009-11-18T05:57:10+09:00
Subject: Re: Trajectories

On Sun, Nov 15, 2009 at 8:22 PM, Eleanor McHugh
<eleanor@games-with-brains.com> wrote:
> On 16 Nov 2009, at 01:12, Todd Benson wrote:
>>
>> On Sun, Nov 15, 2009 at 5:32 PM, Eleanor McHugh
> This isn't an issue of chaotic behaviour (that has a very fixed meaning
> mathematically) but of unnoticeable error.

I'm talking about result.  A small difference in initial conditions
causes big problem.

> The difference between 1e10-13
> and 2e10-13 matters a lot when working on a system which needs to be
> accurate to a resolution of 1e10-14 but not when working to a resolution of
> 1e10-4. The additional nine decimal places tell us nothing meaningful in
> this latter case as we'll still end up rounding the result to zero.

Right, sort of.  See below.

> That's not a bug but a fundamental outcome of the nature of binary coded
> non-integral numbers. Many rational non-integral numbers cannot be expressed
> accurately in binary representations, whilst binary coded decimal brings a
> whole host of other problems: lower information density, higher memory
> usage, and heavier processing load. BCD also does nothing to resolve the
> problem of how to represent irrational numbers such as ð.

Ok, not a bug.  Just makes you want to scream at somebody (yes the
game thing happened to me recently).  The binary inaccuracy thing has
been beaten to death on this list, but you're right.

> The imprecision can indeed stack for complex ballistics systems, depending
> on the complexity of the forces involved. However to the extent of the
> precision chosen for performing these calculations the resultant inaccuracy
> is irrelevant.

I agree.  I thought we were talking about a simple game.  In any case,
small perversions in data (i.e. rounding beforehand) can cause certain
systems to go haywire.  If you stay, for example, at 1e10-4 from the
beginning, you could end up with errors several orders of magnitude
higher, so the "just rounding to zero anyway" argument doesn't make
sense.  This probably doesn't apply so much in a simple parabolic
equation, but the equation is still nonlinear, so one must be terribly
wary.  Error analysis class was way too long ago, otherwise I'd give
an example.  If somebody wants to step in, please do.

>
> http://en.wikipedia.org/wiki/Accuracy_and_precision explains all of this in
> reasonable detail.

Ah, yes, the first thing you learn when studying engineering.  I'm
guessing the only thing affected in the result would be accuracy.
Just a guess.

Todd