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MathGroup Archive 2005

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Re: FullSimplify on Hyperbolic Functions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg53394] Re: [mg53383] FullSimplify on Hyperbolic Functions
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Sun, 9 Jan 2005 23:03:52 -0500 (EST)
  • References: <BE06D85D.C981%carlos@titan.colorado.edu>
  • Sender: owner-wri-mathgroup at wolfram.com

On 9 Jan 2005, at 20:47, Carlos Felippa wrote:

> on 1/9/05 1:46 AM, Andrzej Kozlowski at akoz at mimuw.edu.pl wrote:
>
>> The reasons why some of these do not work have already been explained
>> before: basically they are that FullSimplify when using certain
>> transformations functions, cannot use assumptions during the
>> intermediate steps and also that certain transformation functions only
>> "proceed in one direction". In particular this means that you often
>> need to use explicitly ComplexExpand or TrigToExp with FullSimplify.
>> This is one of such cases:
>>
>> In[1]:=
>> r1 = ArcCosh[1 + b^2/2];
>> r2 = 2*ArcSinh[b/2];
>>
>> In[3]:=
>> FullSimplify[ComplexExpand[
>> r1 - r2], b > 0]
>>
>> Out[3]=
>> 0
>>
>> However, this is more tricky:
>>
>> In[4]:=
>> FullSimplify[ComplexExpand[
>> r1^2 - r2^2], b > 0]
>>
>> Out[4]=
>> -4*Log[(1/2)*(b + Sqrt[
>> 4 + b^2])]^2 +
>> Log[(1/2)*(2 +
>> b*(b + Sqrt[4 + b^2]))]^2
>>
>>
>> The problem seems to be again that FullSimplify does not try certain
>> transformations that go in the "opposite direction" to what it 
>> normally
>> does, but without which it can't see the cancellations. You can make 
>> it
>> see them by forcing expansion of terms:
>>
>>
>> FullSimplify[FunctionExpand[ComplexExpand[r1^2-r2^2],b>0],b>0]
>>
>> 0
>>
>> This is actually an interesting case because it shows a use of
>> FunctionExpand with assumptions in cases where FullSimplify with
>> assumptions is alone not sufficient. Also note that ComplexExpand
>> performs better than TrigToExp here.
>>
>> Of course if you make the expression even more complicated it will get
>> harder and harder for Mathematica to reduce it and at some point it
>> will become completely impossible. This isn't really surprising, is 
>> it?
>>
> True.  But the expression I sent is fairly trivial.  On a scale
> of 0-10, its a 2.  Seems like humans have nothing to fear yet.
>>>
>

Certainly. Also it means you should use computers for what computers 
are good at and human intelligence for what it is good at. 
"Simplification" of expressions is not really algorithmic but requires 
judgement and computers are not very good at using judgement.
This is example maybe fairly trivial from the human viewpoint but it is 
far from trivial from the algorithmic viewpoint. Before the various 
parts of the expression cancel out and thus become simpler they have to 
first be made more "complex". In fact the above example can be done 
even without FunctionExpand, the simpler ExpandAll (without any 
assumptions) works just as well:


FullSimplify[ExpandAll[ComplexExpand[r1^2-r2^2]],b>0]


0

Of course ExpandAll and FunctionExpand do not belong to the usual 
transformation functions used by FullSimplify and for good reason, 
since expanding generally makes things more complex. In most cases 
therefore applying ExpandAll would result in, at best, a substantial 
waste of time. But in this kind of case it seems to be the only way to 
obtain the required cancellation. In such a situation a human being 
will use judgement and intuition that tells one to try expanding the 
expression to see if cancellations might occur. But since a computer 
program can't use intuition or judgement it has to either to include 
ExpandAll always among the transformation functions, which seems like a 
very bad idea, or in some other algorithmic way recognize those cases 
when expanding is useful and those when it will be counterproductive. 
Of course if you can describe such an algorithmic criterion with 
acceptable complexity I am sure that Adam Strzebonski will consider 
using it in Simplify.

Andrzej Kozlowski


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