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MathGroup Archive 2006

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Re: Mathematica bugs?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67323] Re: [mg67301] Mathematica bugs?
  • From: "Carl K. Woll" <carlw at wolfram.com>
  • Date: Sun, 18 Jun 2006 05:13:17 -0400 (EDT)
  • References: <200606170836.EAA27990@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Yaroslav Bulatov wrote:
> When I run the following line
> x = Pi/4; For[i = 0, i < 56, i += 1, x = 2*Abs[x - 1/2]]; N[x]
> I get
> 3.
> 
> But x should always stay between 0 and 1, why do I get 3?
> 
> Also
> x = Pi/4; For[i = 0, i < 50, i += 1, x = 2*Abs[x - 1/2]]; N[Log[x]]
> gives me
> Indeterminate
> 
> How can I get an indeterminate here?

Look at x instead of N[x] (i use i<5 instead):

In[14]:= x = Pi/4; For[i = 0, i < 5, i += 1, x = 2*Abs[x - 1/2]]; {x, N[x]}

Out[14]= {2 (1/2 - 2 (-1/2 + 2 (1/2 - 2 (-1/2 + 2 (-1/2 + Ï?/4))))), 
0.132741}

You see that Mathematica is smart enough to figure out that x-1/2 is 
positive, so that the Abs is unnecessary. On the other hand, Mathematica 
is not expanding x out, so x becomes a bigger and bigger expression as i 
increases. Eventually, the expression involves so many multiplications 
and subtractions that applying N to x experiences catastrophic numerical 
cancellations.

The solution is simple. Use extended precision numbers:

In[16]:= x = Pi/4; For[i = 0, i < 56, i += 1, x = 2*Abs[x - 1/2]]; N[x, 20]

Out[16]= 0.79387245267980382900

You can even use extended precision numbers with only 2 digits of precision:

In[17]:= x = Pi/4; For[i = 0, i < 56, i += 1, x = 2*Abs[x - 1/2]]; N[x, 2]

Out[17]= 0.79

Carl Woll
Wolfram Research


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