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Re: Mathematica bugs?

• To: mathgroup at smc.vnet.net
• Subject: [mg67336] Re: Mathematica bugs?
• From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
• Date: Sun, 18 Jun 2006 05:13:45 -0400 (EDT)
• Organization: The Open University, Milton Keynes, UK
• References: <e70f29$rf4$1@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?

No bug here, just numerical imprecision. Have a look at the 
*Numerical-Precision Tracking (NPT)* entry from the _Technology Guide_ 
at http://wolfram.com/technology/guide/precisiontracking.html

To get the right answer, increase the required precision [1] as in

In[27]:=
x = Pi/4;
For[i = 0, i < 56, i += 1, x = 2*Abs[x - 1/2]];
   N[x, 20]

Out[28]=
0.79387245267980382900

The following expressions may help you to better understand what's going 
on (machine precision vs. arbitrary precision):

In[29]:=
NestList[2Abs[#-1/2]&, Pi/4, 56]//N

Out[29]=
{0.785398, 0.570796, 0.141593, 0.716815, 0.433629,

   0.132741, 0.734518, 0.469035, 0.0619298, 0.87614,

   0.752281, 0.504561, 0.00912272, 0.981755, 0.963509,

   0.927018, 0.854036, 0.708073, 0.416146, 0.167709,

   0.664583, 0.329165, 0.341669, 0.316661, 0.366678,

   0.266645, 0.466711, 0.0665783, 0.866843, 0.733687,

   0.467374, 0.0652523, 0.869495, 0.738991, 0.477982,

   0.0440369, 0.911926, 0.823853, 0.647705, 0.29541,

   0.40918, 0.181641, 0.636719, 0.273438, 0.453125,

   0.09375, 0.8125, 0.625, 0.25, 0.5, 0., 1., 1., 1.,

   1., -1., 3.}

In[30]:=
NestList[2Abs[#-1/2]&, Pi/4, 56]//N[#,20]&

Out[30]=
{0.78539816339744830962, 0.57079632679489661923,

   0.14159265358979323846, 0.71681469282041352307,

   0.43362938564082704615, 0.13274122871834590770,

   0.73451754256330818460, 0.46903508512661636920,

   0.061929829746767261609, 0.87614034050646547678,

   0.75228068101293095356, 0.50456136202586190713,

   0.0091227240517238142532, 0.98175455189655237149,

   0.96350910379310474299, 0.92701820758620948597,

   0.85403641517241897195, 0.70807283034483794390,

   0.41614566068967588780, 0.16770867862064822441,

   0.66458264275870355119, 0.32916528551740710237,

   0.34166942896518579525, 0.31666114206962840950,

   0.36667771586074318101, 0.26664456827851363799,

   0.46671086344297272403, 0.066578273114054551945,

   0.86684345377189089611, 0.73368690754378179222,

   0.46737381508756358444, 0.065252369824872831112,

   0.86949526035025433778, 0.73899052070050867555,

   0.47798104140101735110, 0.044037917197965297793,

   0.91192416560406940441, 0.82384833120813880883,

   0.64769666241627761766, 0.29539332483255523532,

   0.40921335033488952936, 0.18157329933022094128,

   0.63685340133955811744, 0.27370680267911623488,

   0.45258639464176753023, 0.094827210716464939537,

   0.81034557856707012093, 0.62069115713414024185,

   0.24138231426828048371, 0.51723537146343903259,

   0.034470742926878065172, 0.93105851414624386966,

   0.86211702829248773931, 0.72423405658497547863,

   0.44846811316995095725, 0.10306377366009808550,

   0.79387245267980382900}
> 
> 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?
> 

In[34]:=
x = Pi/4;
For[i = 0, i < 50, i += 1, x = 2*Abs[x - 1/2]];
   N[Log[x],20]

Out[36]=
-3.3676443458316025909

Regards,
Jean-Marc

1. http://documents.wolfram.com/mathematica/functions/N