Re: Simplify 0/0 to 1?
- To: mathgroup at smc.vnet.net
- Subject: [mg77516] Re: Simplify 0/0 to 1?
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Mon, 11 Jun 2007 04:25:58 -0400 (EDT)
- References: <934147.40360.qm@web43135.mail.sp1.yahoo.com> <2DFED10A-AAB0-4649-A747-93E238C30B75@mimuw.edu.pl> <B338F0C9-E8DF-405A-802C-C1820700EB83@mimuw.edu.pl>
I need to correct somehtin gin my post below. The three argument
usage of If in the examples below will lead to serious trouble; the
correct way is to use four arguments:
$Pre = (If[Simplify[Denominator[#]] == 0, Print["Zero denominator!"],
#, #] &);
and
$Pre = If[N[Denominator[#1], 20] == 0, Print["Probably zero
denominator!"], #1, #] &;
Andrzej Kozlowski
On 11 Jun 2007, at 10:31, Andrzej Kozlowski wrote:
> In fact, if you are really concerned with this, you could do
> something like:
>
> $Pre = (If[Simplify[Denominator[#]] == 0, Print["Zero
> denominator!"], #] &);
>
> You will then get:
>
> o = (Log[2]*Cos[Pi/12] - Log[2]*Sin[Pi/12] - 2*Cos[Pi/12] + 2*Sin
> [Pi/12] +
> Sqrt[2] + 2*Log[Cos[Pi/12] - Sin[Pi/12]]*Cos[Pi/12] -
> 2*Log[Cos[Pi/12] - Sin[Pi/12]]*Sin[Pi/12])/(Log[2]*Cos[Pi/12] -
> Log[2]*Sin[Pi/12] + 2*Log[Cos[Pi/12] - Sin[Pi/12]]*Cos[Pi/12] -
> 2*Log[Cos[Pi/12] - Sin[Pi/12]]*Sin[Pi/12])
>
> "Zero denominator!"
>
> before you even apply Simplify. Similarly:
>
> (1 - Cos[Pi/23]^2 - Sin[Pi/23]^2)^2/(1 - 2*Cos[Pi/23]^2 +
> Cos[Pi/23]^4 - 2*Sin[Pi/23]^2 + 2*Cos[Pi/23]^2*Sin[Pi/23]^2 +
> Sin[Pi/23]^4)
> "Zero denominator!"
>
> Unfortunately this will produce dramatic slowdowns in any slighlty
> complex algebraic computations (I haven't tried it but I am pretty
> sure). It might be better to replace this with a numerical check,
> but you will have to use extended precision arithemtic, since
> machine precision will often fail to detect zeros. So you could
> instead use:
>
> $Pre =.
> $Pre = If[N[Denominator[#1], 20] == 0,
> Print["Probably zero denominator!"], #1] & ;
> o
> N::meprec:Internal precision limit $MaxExtraPrecision = 50. reached
> while evaluating -((-1+) log(2))/(2 )+((1+) log(2))/(2 )-((-1+) log
> ((1+)/(2 )-<<1>>/(2 <<1>>)))/+((1+) log((1+)/(2 )-(-1+)/(2 )))/. >>
> "Probably zero denominator!"
>
> Peronally I think it is much better to leave thigs as they are,
> keep this in mind and be a little careful.
>
> Andrzej Kozlowski
>
>
> On 10 Jun 2007, at 22:20, Andrzej Kozlowski wrote:
>
>> *This message was transferred with a trial version of CommuniGate
>> (tm) Pro*
>> I would not call this bad performance. In fact this is inevitable,
>> not only in Mathematica but in every CAS that exists today
>> (although not necssarily in the same example). ALl you need to do
>> is to choose numerator and denominator equal to zero but
>> sufficiently complicated and such that the CAS will see that they
>> are equal before it can check that they are both 0. Every CAS will
>> then simplify the expression to 1, because for reason of
>> performance they cancel common factors as soon as they see them.
>> Here is a very trivial example:
>>
>> (1 - Cos[Pi/23]^2 - Sin[Pi/23]^2)/(1 - Cos[Pi/23]^2 - Sin[Pi/23]^2)
>> 1
>>
>> This was turned into 1 by the evaluator withot any Simplify even
>> though both, the numerator and the denominator are clearly zero.
>> You can avoid this by mapping Sinmplify on the numerator and the
>> denominator but you have to use Unevaluated first:
>>
>> Simplify /@
>> Unevaluated[(1 - Cos[Pi/23]^2 - Sin[Pi/23]^2)/(1 - Cos[Pi/23]^2 -
>> Sin[Pi/23]^2)]
>> The same approach will work in your example:
>>
>> Simplify/@o
>>
>> I am convinced that this phenomenon is general will be observed in
>> every CAS, although not necessarily in the same examples. But in
>> every CAS you can construct expressions equal to zero that will be
>> sufficiently complicated for the CAS not to be able to see that
>> they are zero before seeing that they are equal. Here is another
>> example, which is just again based on the Sin^2[a]+Cos^2[a]==1
>> identity, but just made a bit more complicated, so that ti doe not
>> reduce to 1 immediately:
>>
>>
>> Simplify[(1 - Cos[Pi/23]^2 - Sin[Pi/23]^2)^2/(1 - 2*Cos[Pi/23]^2 +
>> Cos[Pi/23]^4 -
>> 2*Sin[Pi/23]^2 + 2*Cos[Pi/23]^2*Sin[Pi/23]^2 + Sin[Pi/23]^4)]
>>
>> 1
>>
>> As you can see again, the numerator and the denominator are both
>> zero but Simplify sees that they are equal before it can see that
>> they are both zero and at that point the Evaluator performs the
>> cancelation. I think for reason of performance all CAS have to
>> work like this in principle.
>>
>> Andrzej Kozlowski
>>
>>
>>
>>
>>
>> On 10 Jun 2007, at 20:57, dimitris anagnostou wrote:
>>
>>>
>>> Hi.
>>> This appeared in another forum as part of a question
>>> what another CAS does.
>>> Just of curiosity I check Mathematica's performance (5.2).
>>> The result was poor!
>>>
>>> Here is the expression
>>>
>>>
>>> In[16]:=
>>> o = (Log[2]*Cos[Pi/12] - Log[2]*Sin[Pi/12] - 2*Cos[Pi/12] + 2*Sin
>>> [Pi/
>>> 12] + Sqrt[2] +
>>> 2*Log[Cos[Pi/12] - Sin[Pi/12]]*Cos[Pi/12] - 2*Log[Cos[Pi/12] -
>>> Sin[Pi/12]]*Sin[Pi/12])/
>>> (Log[2]*Cos[Pi/12] - Log[2]*Sin[Pi/12] + 2*Log[Cos[Pi/12] -
>>> Sin[Pi/
>>> 12]]*Cos[Pi/12] -
>>> 2*Log[Cos[Pi/12] - Sin[Pi/12]]*Sin[Pi/12])
>>>
>>>
>>> Out[16]=
>>> (Sqrt[2] + (-1 + Sqrt[3])/Sqrt[2] - (1 + Sqrt[3])/Sqrt[2] - ((-1 +
>>> Sqrt[3])*Log[2])/(2*Sqrt[2]) +
>>> ((1 + Sqrt[3])*Log[2])/(2*Sqrt[2]) - ((-1 + Sqrt[3])*Log[-((-1 +
>>> Sqrt[3])/(2*Sqrt[2])) + (1 + Sqrt[3])/(2*Sqrt[2])])/
>>> Sqrt[2] + ((1 + Sqrt[3])*Log[-((-1 + Sqrt[3])/(2*Sqrt[2])) +
>>> (1 +
>>> Sqrt[3])/(2*Sqrt[2])])/Sqrt[2])/
>>> (-(((-1 + Sqrt[3])*Log[2])/(2*Sqrt[2])) + ((1 + Sqrt[3])*Log[2])/
>>> (2*Sqrt[2]) -
>>> ((-1 + Sqrt[3])*Log[-((-1 + Sqrt[3])/(2*Sqrt[2])) + (1 + Sqrt
>>> [3])/
>>> (2*Sqrt[2])])/Sqrt[2] +
>>> ((1 + Sqrt[3])*Log[-((-1 + Sqrt[3])/(2*Sqrt[2])) + (1 + Sqrt[3])/
>>> (2*Sqrt[2])])/Sqrt[2])
>>>
>>> Watch now a really bad performance!
>>>
>>> In[17]:=
>>> (Simplify[#1[o]] & ) /@ {Numerator, Denominator}
>>>
>>> Out[17]=
>>> {0, 0}
>>>
>>> That is Mathematica simplifies succesfully both the numerator
>>> and denominator to zero. So, you wonder what goes wrong?
>>>
>>> Try now to simplify the whole expression!
>>>
>>> In[19]:=
>>> Simplify[o]
>>>
>>> Out[19]=
>>> 1
>>>
>>> A very weird result to my opinion!
>>> Simplification of 0/0 to 1?
>>> I think no simplification or some
>>> warning messages would be much better
>>> than 1!
>>>
>>> Note also that
>>>
>>> In[20]:=
>>> RootReduce[o]
>>>
>>> Out[20]=
>>> 1
>>>
>>> Dimitris
>>>
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>>
>