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MathGroup Archive 2003

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Re: A question on interval arithmetic

  • To: mathgroup at smc.vnet.net
  • Subject: [mg43725] Re: A question on interval arithmetic
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Thu, 2 Oct 2003 02:51:21 -0400 (EDT)
  • Organization: The University of Western Australia
  • References: <blcq5l$p5f$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <blcq5l$p5f$1 at smc.vnet.net>,
 Oliver Friedrich <oliver.friedrich at tzm.de> wrote:

> the resistance of 2 resistors in parallel is r1*r2/(r1+r2). Now I want to 
> introduce tolerances in the resistors and ask for the range of resistance 
> of the combination. One may think that e.g
> 
> (r1*r2)/(r1+r2)/.{r1->Interval[{10,20}],r2->Interval[{20,40}]}
> 
> would lead to the correct result, but there's a trap. If I replace the 
> expressions by the intervals, Mathematica evaluates the new expression 
> assuming that all four intervals are independant from each other. And that's not 
> correct. Taken either the minimum or the maximum from a certain interval , 
> Mathematica should stick to that, because it is nonsense to take Min[r1] and 
> Max[r1] within the same expression, r1 can have only one value at a time.
> 
> How can I avoid this problem?

Defining the resistance of 2 resistors in parallel

  R[r1_,r2_] = 1/(1/r1+1/r2)

then, in this simple example, you could just compute the extremal values 
directly:

  {R[10, 20], R[20, 40]}

yielding the minimum and maximum possible values.

A better (completely general) solution is to use the calculus 
(quadrature) of errors:

  error[R_][{r1_,e1_},{r2_,e2_}] = 
   Sqrt[D[R[r1,r2],r1]^2 e1^2+D[R[r1,r2],r2]^2 e2^2]

Then you can compute the range of resistance of the combination by 
entering Interval[{10,20}] as {15,5} etc:

  error[R][{15, 5}, {30, 10}]

Cheers,
Paul

-- 
Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
The University of Western Australia      (CRICOS Provider No 00126G)         
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Crawley WA 6009                      mailto:paul at physics.uwa.edu.au 
AUSTRALIA                            http://physics.uwa.edu.au/~paul



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