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MathGroup Archive 2005

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Re: //N bug, but WHY?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg58655] Re: //N bug, but WHY?
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Tue, 12 Jul 2005 05:21:38 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, England
  • References: <data3n$mec$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

symbio wrote:
> Evaluating (using //N) two exact same expressions, gives wrong answer unless 
> fullsimplify is used first, I spent 2 days on a problem thinking my answer 
> was wrong, but turned out Mathematica 5 was giving me buggy answers, I 
> debugged it to this point, but WHY in the world is this happening?  Please 
> help!!!
> 
> cut and paste below to see the test case:
> 
> In[243]:=
> \!\(\(Cosh[\(43\ \[Pi]\)\/\@2] + \((1 - Cosh[43\ \@2\ \[Pi]])\)\ Csch[
>             43\ \@2\ \[Pi]]\ Sinh[\(43\ \[Pi]\)\/\@2] // FullSimplify\) //
>     N\[IndentingNewLine]
>   Cosh[\(43\ \[Pi]\)\/\@2] + \((1 - Cosh[43\ \@2\ \[Pi]])\)\ Csch[
>           43\ \@2\ \[Pi]]\ Sinh[\(43\ \[Pi]\)\/\@2] // N\)
> Out[243]=
> \!\(6.551787517854307`*^-42\)
> Out[244]=
> \!\(\(-1.9342813113834067`*^25\)\)

No bug and nothing magic here; just the misunderstanding of what 
machine-precision arithmetic is and its pitfalls in some circumstances.

In[1]:=
expr = Cosh[(43*Pi)/Sqrt[2]] + (1 - Cosh[43*Sqrt[2]*Pi])*
     Csch[43*Sqrt[2]*Pi]*Sinh[(43*Pi)/Sqrt[2]]

Out[1]=
Cosh[(43*Pi)/Sqrt[2]] + (1 - Cosh[43*Sqrt[2]*Pi])*Csch[43*Sqrt[2]*Pi]*
    Sinh[(43*Pi)/Sqrt[2]]

In[2]:=
exprfs = FullSimplify[expr]

Out[2]=
Sech[(43*Pi)/Sqrt[2]]

Both expressions do not seem to be "two exact same expressions." Let try 
to expand the first one to see what happens:

In[3]:=
TrigExpand[expr]

Out[3]=
1/(Cosh[Pi/Sqrt[2]]^43 + 903*Cosh[Pi/Sqrt[2]]^41*Sinh[Pi/Sqrt[2]]^2 +
    123410*Cosh[Pi/Sqrt[2]]^39*Sinh[Pi/Sqrt[2]]^4 +
    6096454*Cosh[Pi/Sqrt[2]]^37*Sinh[Pi/Sqrt[2]]^6 +
    145008513*Cosh[Pi/Sqrt[2]]^35*Sinh[Pi/Sqrt[2]]^8 +
    1917334783*Cosh[Pi/Sqrt[2]]^33*Sinh[Pi/Sqrt[2]]^10 +
    15338678264*Cosh[Pi/Sqrt[2]]^31*Sinh[Pi/Sqrt[2]]^12 +
    78378960360*Cosh[Pi/Sqrt[2]]^29*Sinh[Pi/Sqrt[2]]^14 +
    265182149218*Cosh[Pi/Sqrt[2]]^27*Sinh[Pi/Sqrt[2]]^16 +
    608359048206*Cosh[Pi/Sqrt[2]]^25*Sinh[Pi/Sqrt[2]]^18 +
    960566918220*Cosh[Pi/Sqrt[2]]^23*Sinh[Pi/Sqrt[2]]^20 +
    1052049481860*Cosh[Pi/Sqrt[2]]^21*Sinh[Pi/Sqrt[2]]^22 +
    800472431850*Cosh[Pi/Sqrt[2]]^19*Sinh[Pi/Sqrt[2]]^24 +
    421171648758*Cosh[Pi/Sqrt[2]]^17*Sinh[Pi/Sqrt[2]]^26 +
    151532656696*Cosh[Pi/Sqrt[2]]^15*Sinh[Pi/Sqrt[2]]^28 +
    36576848168*Cosh[Pi/Sqrt[2]]^13*Sinh[Pi/Sqrt[2]]^30 +
    5752004349*Cosh[Pi/Sqrt[2]]^11*Sinh[Pi/Sqrt[2]]^32 +
    563921995*Cosh[Pi/Sqrt[2]]^9*Sinh[Pi/Sqrt[2]]^34 +
    32224114*Cosh[Pi/Sqrt[2]]^7*Sinh[Pi/Sqrt[2]]^36 +
    962598*Cosh[Pi/Sqrt[2]]^5*Sinh[Pi/Sqrt[2]]^38 +
    12341*Cosh[Pi/Sqrt[2]]^3*Sinh[Pi/Sqrt[2]]^40 +
    43*Cosh[Pi/Sqrt[2]]*Sinh[Pi/Sqrt[2]]^42)

Wow! Things are getting worse :-) except if the second expression can be 
expanded in a similar way. Let's check

In[4]:=
TrigExpand[expr] == TrigExpand[exprfs]

Out[4]=
True

Good. Now let's check that the expressions by themselves are equal:

In[5]:=
expr == exprfs

 From In[5]:=
\!\(\*
   RowBox[{\(N::"meprec"\),
    ":", "\<\"Internal precision
       limit $MaxExtraPrecision = \\!\\(49.99999999999999`\\) reached 
while \
evaluating \\!\\(\\(\\(Cosh[\\(\\(\\(43\\\\ Ï?\\)\\/\\@2\\)\\)]\\)\\) - \
\\(\\(Sech[\\(\\(\\(43\\\\ Ï?\\)\\/\\@2\\)\\)]\\)\\) + \\(\\(\\(\\((1 - \
\\(\\(Cosh[\\(\\(43\\\\ \\@2\\\\ Ï?\\)\\)]\\)\\))\\)\\)\\\\ \
\\(\\(Csch[\\(\\(43\\\\ \\@2\\\\ Ï?\\)\\)]\\)\\)\\\\ \
\\(\\(Sinh[\\(\\(\\(43\\\\ Ï?\\)\\/\\@2\\)\\)]\\)\\)\\)\\)\\). \
\\!\\(\\*ButtonBox[\\\"Moreâ?¦\\\", ButtonStyle->\\\"RefGuideLinkText\\\", \
ButtonFrame->None, ButtonData:>\\\"General::meprec\\\"]\\)\"\>"}]\)

Out[5]=
Cosh[(43*Pi)/Sqrt[2]] + (1 - Cosh[43*Sqrt[2]*Pi])*Csch[43*Sqrt[2]*Pi]*
     Sinh[(43*Pi)/Sqrt[2]] == Sech[(43*Pi)/Sqrt[2]]

Okay. Mathematica is not able to tell us whether both expressions are 
equal; however, the warning message gives us a crucial clue. Thus we now 
try to use  50 digits, say.

In[6]:=
N[expr, 50]

 From In[6]:=
\!\(\*
   RowBox[{\(N::"meprec"\),
    ":", "\<\"Internal precision
       limit $MaxExtraPrecision = \\!\\(49.99999999999999`\\) reached 
while \
evaluating \\!\\(\\(\\(Cosh[\\(\\(\\(43\\\\ Ï?\\)\\/\\@2\\)\\)]\\)\\) + 
\\(\\(\
\\(\\((1 - \\(\\(Cosh[\\(\\(43\\\\ \\@2\\\\ Ï?\\)\\)]\\)\\))\\)\\)\\\\ \
\\(\\(Csch[\\(\\(43\\\\ \\@2\\\\ Ï?\\)\\)]\\)\\)\\\\ \
\\(\\(Sinh[\\(\\(\\(43\\\\ Ï?\\)\\/\\@2\\)\\)]\\)\\)\\)\\)\\). \
\\!\\(\\*ButtonBox[\\\"Moreâ?¦\\\", ButtonStyle->\\\"RefGuideLinkText\\\", \
ButtonFrame->None, ButtonData:>\\\"General::meprec\\\"]\\)\"\>"}]\)

Out[6]=
6.551787517854344014050058`14.31640268976547*^-42

We still get a warning message but things are getting definitely better. 
Having a look at the specified documentation, we try

In[7]:=
Block[{$MaxExtraPrecision = 1000}, N[expr, 50]]

Out[7]=
6.55178751785434401405005796823578794510303347370361442804220253`50.*^\
-42

In[8]:=
N[exprfs, 50]

Out[8]=
6.55178751785434401405005796823578794510303347370361442804220253`50.*^\
-42

and we can see that both values are equal (up to 50 digits in our 
example). We have just done is using arbitrary-precision arithmetic 
(safe but "slow") rather than machine-size precision arithmetic (fast 
but sometimes "surprising", hardware dependent).

Regards,
/J.M.


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