Re: Numerical accuracy/precision - this is a bug or

*To*: mathgroup at smc.vnet.net*Subject*: [mg120265] Re: Numerical accuracy/precision - this is a bug or*From*: "Christoph Lhotka" <christoph.lhotka at univie.ac.at>*Date*: Fri, 15 Jul 2011 05:15:14 -0400 (EDT)

Dear Group! I do not agree to arguments based on SetPrecision, since the function modifies the numbers (it adds binary 0 to fill the missing binary digits as stated in the documentation) it does not show us what a and b are. It is no suprise for me that in base 10 the numbers a and b are now different (and SameQ qill return False). If we want to see the actual form of a and b I would rather use FullForm: a = 1.100000000000000000000000000; b = 1.1; FullForm/@{a,b} {1.1000000000000000000000000000000000000000000000000000000001`27.041392685158225,1.1`} As we can see the first number is accurate up to the 27th digit while the second is accurate up to machine precision. Well but they are still different! First I was surprised that SameQ yields True but I guess that it has something to do with Mathematica's principle to return results correct to the lowest precision found in the expression. So to my opinion Mathematica does the following: 1) first convert all numbers to numbers of the lowest precision found in the expression, 2) follow the usual evaluation chain. Am I correct? If so, SameQ yields True, since up to machine precision both numbers are the same expression. In another response to my post I learned that it is not a good idea to overwrite the definition of Equal. As a conclusion I would implement the following identity: myEqual[a_,b__]:=And@@{SameQ@@Head/@{a,b},SameQ@@Precision/@{a,b},SameQ@@{a,b},Equal@@{a,b}} a_\[TildeEqual]b__:=myEqual[a,b] which also handles the issue with the comparison of different precision numbers: In[62]:= a\[TildeEqual]a Out[62]= True In[64]:= a\[TildeEqual]b Out[64]= False Of course the definition can be modified to your needs... Best, Christoph On 15/07/2011 10:08, James Stein wrote: > Richard, thanks for pointing this out. > >From what you say, it seems that the documentation for SameQ is incorrect. > > On Thu, Jul 14, 2011 at 6:21 PM, Richard Fateman > <fateman at eecs.berkeley.edu>wrote: > >> On 7/14/2011 6:22 AM, Christoph Lhotka wrote: >>> Dear group! >>> >> .... >> >>> My question is: how dangerous is it to overwrite such a fundamental >>> function as Equal? >>> >>> Best, >>> >>> Christoph >>> >> An excellent question. Anyone who runs Mathematica in serious >> applications could try it >> and report back. >> >> Unfortunately, SameQ doesn't operate the way I suspect you think it does. >> >> Consider a = 1.100000000000000000000000000; b=1.1 >> >> SameQ[a,b] is True. >> >> Yet a and b are different. You can see this if you do >> SetPrecision[a,100] and SetPrecision[b,100]. >> >> 1.10000000000000000000000000000000000000117549435082228750796873653722\ >> 2245677818665556772087521508752 >> >> 1.10000000000000008881784197001252323389053344726562500000000000000000\ >> 0000000000000000000000000000000 >> >> RJF >> >> >