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Re: Interval arithmetic and Dependency problems

  • To: mathgroup at smc.vnet.net
  • Subject: [mg95500] Re: Interval arithmetic and Dependency problems
  • From: "David W. Cantrell" <DWCantrell at sigmaxi.net>
  • Date: Tue, 20 Jan 2009 05:46:44 -0500 (EST)
  • References: <gl1bng$998$1@smc.vnet.net>

magma <maderri2 at gmail.com> wrote:
> I just started to have a look at Mathematica capabilities in Interval
> arithmetic.
>
> It seems that Mathematica does not consider the so called dependency
> problem (see for instance wikipedia on Interval arithmetic for an
> explanation).

The dependency problem is naturally present in interval arithmetic, as you
should have gathered from reading the Wikipedia article. It is not
Mathematica, but rather the _user_ of Mathematica, that needs to consider
dependency when using interval arithmetic.

> For example evaluate
>
> f[x_] := x^2 + x
> f[Interval[{-1, 1}]]
>
> and you get
>
> Interval[{-1, 2}]
>
> which is not really very good.
> The correct or, if you prefer, best result should be
>
> Interval[{-1/4,2}]
>
> since -1/4 and 2 are really the min and max values of f over the given
> interval.

So, just as the Wikipedia article mentions, you merely need to rewrite your
function:

f[x_] := (x + 1/2)^2 - 1/4

Then, using Mathematica, f[Interval[{-1, 1}]] yields Interval[{-1/4, 2}].

David


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