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MathGroup Archive 1996

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Machine arithmetics Q?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg3657] Machine arithmetics Q?
  • From: "Arturas Acus" <acus at itpa.lt>
  • Date: Tue, 2 Apr 1996 21:55:09 -0500
  • Comments: Authenticated sender is <acus@mserv.itpa.lt>
  • Sender: owner-wri-mathgroup at wolfram.com

Hello,

It is known, that following two formulas 
give the same result. 

sinusas[i_]:=sinusas[i]=N[Sin[2^i*ArcSin[Sqrt[x0]]]^2,prec];
iter[i_]:=iter[i]=4*iter[i-1]*(1-iter[i-1]);

I want to demonstrate, that we must calculate correct.
That is, if we use $MachinePrecision  numbers after some 
steps we will get random numbers:

prec=5;maximum=100;
iter[0]=x0=N[1/3,prec];
deltalist=ListPlot[Table[
N[(sinusas[i]-iter[i]),prec],{i,maximum}]]

This result is ok., because  we 
have positive and negative differences in calculations.

Now I want to show, that if we will use 20 digits, and 
machine arithmetic (Not Big Numbers Arithmetic !)
we will get chaos later. Therefore I remove Big Numbers Arithmetic:

$MinPrecision=$MaxPrecision=20;
prec=20;

and perform the same calculations. What I get is quite different!
After some iterations I get only negative deviations!
Can anybody explain why this happens?
I use Mathematica 2.2.1 for Windows.

Thanks in advise 
                  Arturas Acus

E-mail: acus at itpal.lt

Institute of Theoretical Physics and Astronomy,
Gostauto 12, 2600, Vilnius, Lithuania.



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