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From: v104n55k@ubvms.cc.buffalo.edu (David I Schwartz)
Subject: Re: Question Re: Interval Mathematics
Message-ID: <CwIADJ.4xM@acsu.buffalo.edu>
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Date: Thu, 22 Sep 1994 00:52:00 GMT
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Xref: glinda.oz.cs.cmu.edu sci.math:81052 comp.constraints:293

In article <35n3mr$f4b@babbage.ece.uc.edu>, wbradley@potato.ece.uc.edu (William Bradley) writes...
> 
>I'm looking into interval analysis.  One problem I've come across is
>that of interval evaluation of real expressions.  That is, for a function
>f and its interval equivalent F:
> 
>	F(X,Y...) = { z | z = f(x,y...), x \in X, y \in Y, ... }
> 
>This is no big deal for lots of functions, except where a variable is
>used more than once in the expression for f(X,Y...).  I've seen this
>called the "dependency" problem (by Hansen, 1992).
> 
>For example, clearly anything minus itself is zero.  However, if X = [2,5],
> 
>	X - X = [2,5] - [2,5] = [-3,3]
> 
>Granted, this *includes* the correct result, but I'd like something better.
> 
>Hansen briefly mentions two ways to approach this.  The first is one of
>algebraic simplification; the second is only useful in linear systems.
>Neither of these are really applicable in my application.
> 
>Is there any easy way around this problem short of an algebraic simplifier
>(that is, something that can recognize X - X and replace it with [0,0])?
> 

It depends on your application. From IA (Interval Analysis) theory, any 
value contained in set X is possible; thus, the statement "X-X" should not
necessarily imply zero. There are some short cuts:

1) redefine IA (see some works on Fuzzy Arithmetic for alternate rules)
   - somewhat drastic, but you can define any rules you prefer, but you
     must justify and develop your own separate algebra
2) see Alefeld 83 for review on basic IA in chap 1 (he refers to a paper
   by Berti that deals with ax + b = c problems - sometimes a "tighter"
   ("algebraic solution") is possible)

   For example, given:
   [2,4] x = [1,8]

   x = [1/2,2] as "algebraic solution" ([2,4][1/2,2] = [1,8])
   x = [1/4,4] by interval division    (x is in [1,8]/[2,4])

   (Alefeld gives Ratschek's conditions for this.)

3)  Moore 66 and others describe a centered IA (the center of intervals
    is computed as well)

I suppose there are other ways for dealing with this. I'm curious to see what
others have to say. Personally, I am skeptical of limiting such "X-X" cases
to single values as this defeats the purpose of IA.

later,
Dave

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