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MathGroup Archive 2004

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Re: Re: Re: FindRoot cannot find obvious solution

  • To: mathgroup at smc.vnet.net
  • Subject: [mg48129] Re: [mg48092] Re: [mg48085] Re: FindRoot cannot find obvious solution
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Fri, 14 May 2004 00:12:27 -0400 (EDT)
  • References: <200404270847.EAA18892@smc.vnet.net> <c6o3lc$cd0$1@smc.vnet.net> <c6qags$s56$1@smc.vnet.net> <200405080524.BAA11690@smc.vnet.net> <c7klcs$2kn$1@smc.vnet.net> <200405110920.FAA28320@smc.vnet.net> <200405130408.AAA26661@smc.vnet.net> <40A38589.6080707@wolfram.com>
  • Sender: owner-wri-mathgroup at wolfram.com

On 13 May 2004, at 23:26, Daniel Lichtblau wrote:
>
> The issue of 1`2==2.2 is a bit dicey. I believe the most useful 
> interpretation is roughly as Andrzej suggests, that the model breaks 
> down at sufficiently low precision. Yes, there are "contradictions" is 
> the logic that lead to the equality above, but to me they are not 
> terribly compelling.
>
>
> Daniel
>
>
At least in this respect Mathemaitca's behaviour agrees with the 
documentation: since

RealDigits[1`2,2]

{{1,0,0,0,0,0,0},1}

has only 7 digits it does "differ in at most last eight binary digits" 
from 2.

This is a bit more tricky

1.`2 == 0


True

Sin[1`2]==0

False

The key point is clearly the fact that here Sin slightly increases 
precision:


Precision[Sin[1`2]]


2.1924

Looking at RealDigits[Sin[1`2],2] is not quite convincing, since 
Mathematica still returns only 7 digits:


RealDigits[Sin[1`2],2]


{{1,1,0,1,1,0,0},0}

but I suspect that the statement in the documentation about "last eight 
binary digits" is also only approximate and that this is actually a 
border line case (just over the border line).

In any case, these are clearly pathologies of significance arithmetic 
and without importance.




Andrzej Kozlowski
Chiba, Japan
http://www.mimuw.edu.pl/~akoz/


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