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MathGroup Archive 2004

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Re: Uniform design

  • To: mathgroup at smc.vnet.net
  • Subject: [mg48185] Re: Uniform design
  • From: ab_def at prontomail.com (Maxim)
  • Date: Mon, 17 May 2004 03:21:53 -0400 (EDT)
  • References: <c7nnc7$dm5$1@smc.vnet.net> <200405130408.AAA26737@smc.vnet.net> <c81has$4rj$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote in message news:<c81has$4rj$1 at smc.vnet.net>...
> On 13 May 2004, at 13:08, Maxim wrote:
> 
> > Also, it's strange that Solve accepts intervals (Mathematica Help for
> > Interval even gives such an example), but doesn't really support them:
> >
> > In[7]:=
> > Solve[1/(x - 1) == Interval[{-1, 1}]]
> >
> > Out[7]=
> > {{x -> Interval[{-Infinity, Infinity}]}}
> >
> > Not much point in treating this equation as Solve[1/(x-1)==a,x] and
> > giving incorrect result.
> >
> I agree that it seems strange that this sort of thing was included in 
> the help browser, without additional comment,  for it can certainly 
> only be misleading. Interval arithmetic is strange and does not obey 
> usual rules:
> 
> 1 + 1/Interval[{-1, 1}]
> 
> Interval[{-Infinity, 0}, {2, Infinity}]
> 
> and
> 
> (1 + Interval[{-1, 1}])/Interval[{-1, 1}]
> 
> Interval[{-Infinity, Infinity}]
> 
> This means that the answer returned by Solve will depend on how you 
> choose to write your equation:
> 
> 
> Solve[1/(x - 1) == Interval[{-1, 1}]]
> 
> 
> {{x -> Interval[{-Infinity, Infinity}]}}
> 
> 
> Solve[x - 1 == 1/Interval[{-1, 1}]]
> 
> {{x -> Interval[{-Infinity, 0}, {2, Infinity}]}}
> 
> 
> (What is actually weird is that
> 
> 
> Solve[1/(x - 1) == Interval[{-1, 1}], x]
> 
> {}
> 
> while
> 
> 
> Solve[x - 1 == 1/Interval[{-1, 1}], x]
> 
> {{x -> Interval[{-Infinity, 0}, {2, Infinity}]}})
> 
> 
> Whether the original answer should be considered wrong or only 
> excessively "pessimistic" depends on the context. The usual context in 
> which  interval arithmetic is used is for error estimation, where it is 
> most important that it should  not return an interval smaller than the 
> correct one and at least in this case it does not.
> 
> 
> 
> Andrzej Kozlowski
> Chiba, Japan
> http://www.mimuw.edu.pl/~akoz/

The only problem with that kind of explanation is that it's made with hindsight.

In[1]:=
Solve[1/(x - 1) == Interval[{-1, 0}]]

Out[1]=
{{x -> -1}}

What happened to "pessimistic interval" here?

Maxim Rytin
m.r at prontomail.com


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