[Axiom-developer] address@hidden: Re: [Frink] Interval Arithmetic in Practice]

[root]

I've long believed (and even did thesis work) that interval arithmetic
is the best way to implement provisos. For purely arithmetic results
it can generate interesting results. For symbolic results the answers
are even more interesting.


Frink played with interval arithmetic and got some interesting results:


------- Start of forwarded message -------
To: address@hidden
From: Scott Kirkwood <address@hidden>
Date: Tue, 14 Feb 2006 14:11:04 -0300
Subject: Re: [Frink] Interval Arithmetic in Practice


Alan,
Thought you might be interested in this link about Interval Arithmetic
 http://research.sun.com/minds/2004-0527/

Scott

On 9/6/05, Alan Eliasen <address@hidden> wrote:
>
>
>    I've finally finished implementing the two-argument version of
> arctan[x,y] in Frink to work with interval arithmetic.  This is an
> interesting, subtle, and quite complex function when applied to intervals.
>
>    To inaugurate this function, I decided to test it against real-world
> problems--in this case, a geocaching problem.  If you're not familiar
> with geocaching, it's a growing hobby in which you find hidden sites
> using a GPS receiver.  See http://www.geocaching.com/ .
>
>    I've been trying to solve geocaching puzzles purely geometricallly
> using Frink and spherical geometry.  In this case, I attacked the "Ball
> Bearing" cache:
>
>    http://tinyurl.com/b2thv
>
>    This puzzle asks you to triangulate from two points over a distance
> of several miles to find the solution.  However, the bearings were only
> given to the closest decimal degree, and I wanted to see how much
> uncertainty this gave in the answer.  I had already written a Frink
> program to do the calculations:
>
> http://futureboy.us/fsp/colorize.fsp?fileName=BallBearing.frink
>
> which uses my extensive sun/moon/navigation library:
>
> http://futureboy.us/fsp/colorize.fsp?fileName=sun.frink
>
>    Note that not a single calculation was ever written with intervals in
> mind.  The equations were spherical geometry equations, taken directly
> from texts, containing nests of trigonometric functions.  (It gets even
> worse when you take into account the fact that the earth is an oblate
> spheroid.)  Performing traditional error analysis on these equations
> would be quite painful.
>
>    However, using the magnificient tools of interval arithmetic, it's
> trivial.  In my Frink program, all I did was change lines like:
>
> b7 = 334 degrees
>   to
> b7 = new interval[333.5, 334, 334.5] degrees
>
>    to reflect the fact that the bearing was only known to the closest
> degree.  Run the program again, and Frink automagically tracks these
> error bounds through *all* of those calculations, and produces a
> worst-case figure for the uncertainty in the target!  It literally
> couldn't be easier.
>
>    I could readily see, given the uncertainties, that the target could
> lie in a square (or ellipse) about 0.14 miles on a side.
>
>    It's this sort of amazing, effortless power that interval arithmetic
> gives you that makes it so compelling.  Using Frink's near-transparent
> interval arithmetic makes it easy to apply this sort of analysis to
> nests of formerly intractable equations.  Try it.
>
>    Some geocaching tools powered by Frink:
> http://futureboy.us/fsp/geocaching.fsp
>
> --
>   Alan Eliasen                 |  "It's amazing how much mature wisdom
>   address@hidden       |   resembles being too tired."
>   http://futureboy.homeip.net/ |     -- Robert Heinlein