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MathGroup Archive 2002

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RE: RE: RE: Accuracy and Precision

  • To: mathgroup at smc.vnet.net
  • Subject: [mg37007] RE: [mg36989] RE: [mg36936] RE: Accuracy and Precision
  • From: "DrBob" <drbob at bigfoot.com>
  • Date: Sun, 6 Oct 2002 05:32:51 -0400 (EDT)
  • Reply-to: <drbob at bigfoot.com>
  • Sender: owner-wri-mathgroup at wolfram.com

I suppose my silliness is understandable, in light of all the confusion,
both here and in the Browser on what "significance arithmetic" is, what
"bignums" and "bigfloats" might be, etc.

If many smart people are confused, there's a possibility --- just a
possibility, mind you --- that it isn't entirely their fault.  Yes?  No?

>>like 71 above, or -5 for Accuracy in the example that fooled me), but
it's not a big deal

For people who don't understand it as well as you, yes, it's a big deal.
My purpose in all this is to understand the issue well enough to know
how to proceed.  I think that I do, now, but I doubt everybody on the
list does.

Daniel says this will all be clearer in the next release, and that's
good!

Bobby

-----Original Message-----
From: Andrzej Kozlowski [mailto:andrzej at tuins.ac.jp] 
To: mathgroup at smc.vnet.net
Subject: [mg37007] Re: [mg36989] RE: [mg36936] RE: Accuracy and Precision


On Friday, October 4, 2002, at 06:01 PM, DrBob wrote:

> Here's an even more extreme result:
>
> f = SetAccuracy[333.75*b^6 + a^2*(11*a^2*b^2 - b^6 -
>         121*b^4 - 2) + 5.5*b^8 + a/(2*b), 50];
> a = 77617.; b = 33096.;
> f
> Precision[f]
>
> -1.180591620717411303424`71.0721*^21
> 71
>
> 71.0721 digits of precision?  I don't think so!!

Either I am it altogether or you are just simply beating to death the 
point that in case of machine arithmetic (only!)  Precision and 
Accuracy are purely formal and essentially meaningless.
One can argue whether in this case there is any point of returning any 
value for Precision, or Accuracy (like 71 above, or -5 for Accuracy in 
the example that fooled me), but it's not a big deal and it most 
certainly does not make SetPrecision meaningless. On the contrary, 
SetPrecision is very useful and in fact it is SetPrecision that can 
tell you that the answer above is meaningless:

In[8]:=
f = SetAccuracy[333.75*b^6 + a^2*(11*a^2*b^2 - b^6 -
         121*b^4 - 2) + 5.5*b^8 + a/(2*b), 50];
a=SetPrecision[77617.,$MachinePrecision];  b = SetPrecision[
           33096.,$MachinePrecision];


In[10]:=
{f,Precision[f]}


Out[10]=
{1.19801754103509`0*^19, 0}

I would say this is correct and show that SetPrecision is very useful 
indeed. It tells you (what of course you ought to already know in this 
case anyway) that machine precision will not give you a realiable 
answer in this case. If you know your numbers with a great deal of 
accuracy you can get an accurate answer:

In[24]:=
f = SetAccuracy[333.75*b^6 + a^2*(11*a^2*b^2 - b^6 -
         121*b^4 - 2) + 5.5*b^8 + a/(2*b), 100];
a=SetPrecision[77617.,100];  b = SetPrecision[33096.,100];


In[26]:=
{f, Precision[f]}

Out[26]=
{-0.82739605994682136814116509547981629199903311578438481991\
781484167246798617832`61.2597, 61}

Again you can be pretty sure that you got an accurate answer, provided 
of course your original setting of precision was valid.

Honestly, to say that SetPrecision and SetAccuaracy are useless is one 
of the silliest thing I have read on this list in years.


>
Andrzej Kozlowski
Yokohama, Japan
http://www.mimuw.edu.pl/~akoz/
http://platon.c.u-tokyo.ac.jp/andrzej/





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