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MathGroup Archive 2007

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Re: questions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg77548] Re: questions
  • From: dimitris <dimmechan at yahoo.com>
  • Date: Wed, 13 Jun 2007 07:16:25 -0400 (EDT)
  • References: <200706110824.EAA20819@smc.vnet.net><f4lb09$f7b$1@smc.vnet.net>

Thanks a lot!
As always having finished the reading of your replies
I realized the new and important things I learned!

Dimitris

 /  Andrzej Kozlowski       :
> First, note that in Mathematica 6.0 we have:
>
> ff = Pi*Cos[(1/7)*Pi]*Cos[(2/7)*Pi]*
>      (Cos[(3/7)*Pi]/Sin[Pi*Cos[(1/7)*Pi]*
>         Cos[(2/7)*Pi]*Cos[(3/7)*Pi]]);
>
> FullSimplify[ff]
>
> (1/8)*Pi*Csc[Pi/8]
>
> FunctionExpand[%]
>
> Pi/(4*Sqrt[2 - Sqrt[2]])
>
> Second, simply adding transformations to TransformationFunctions does
> not guarantee that the simplification you desire will be made because
> FullSimplify has to generate by means of the transformation functions
> a chain of expressions, beginning withe the input and ending with the
> desired one, and moreover, there is some limit on how much the
> complexity of generated expressions can increase before they are
> abandoned. (I think for Simplify it is not allowed to increase at
> all). I think that this also explains why in Mathematica 5.2 adding
> FullSimplify to TransformationFunctions works in thi case. What
> happens, I think, is this. Mathematica applies TrigReduce (which is
> one of the TransformationFunctions it uses Automatically)  to ff:
>
> fff = TrigReduce[ff]
>
> (1/4)*(Pi + Pi*Cos[(2*Pi)/7] + Pi*Cos[(4*Pi)/7] +
>     Pi*Cos[(6*Pi)/7])*Csc[Pi/4 + (1/4)*Pi*Cos[(2*Pi)/7] +
>      (1/4)*Pi*Cos[(4*Pi)/7] + (1/4)*Pi*Cos[(6*Pi)/7]]
>
> Now, this has much higher ComplexityFunction than fff:
>
> LeafCount/@{ff,fff}
>
> {41,70}
>
> and would normally be abandoned after applying to it some of the
> other transformation functions. However, one of the
> TransformationFunctions is FullSimplify and when it is applied this
> happens:
>
> FullSimplify[fff]
>
> (1/8)*Pi*Csc[Pi/8]
>
> Now, of course this leaves open the question: why was FullSimplify
> able to simplify this? That I do not know, but I think that more than
> one transformation used by FullSimplify was needed to do this, which
> is why this would not have happened if FullSimplify was not one of
> the TransformationFunctions. In other words, this time it is not the
> question of the transformations that are being used, since
> FullSimplify knows enough transformation to simplify this expression,
> but of the "length of chain of transformed expressions"  used. With
> FullSimplify as a transformation function this chain is in effect
> longer.
>
> So, (assuming that I am right) for me, the interesting question that
> arises from all this is: what has exactly changed in Mathematica 6
> that makes it possible to get the desired answer without adding
> FullSimplify to the TransformationFunctions? There seem to be just
> two possibilities. One is that a new function has been added to
> FullSimplify default transformation functions, or that a new
> "standard form" is being used for certain expressions. The other
> possibility is that the actual way in which FullSimplify works has
> been altered, e.g. perhaps longer "chains of transformations" are
> being allowed. Personally I would put my money on the former.
>
> Andrzej Kozlowski
>
>
>
>
>
>
>
>
> On 11 Jun 2007, at 17:24, dimitris wrote:
>
> > Hello.
> >
> > This appeared recently, but sice there was
> > no response, I make one more attempt.
> >
> > ff = Pi*Cos[1/7*Pi]*Cos[2/7*Pi]*
> >   Cos[3/7*Pi]/Sin[Pi*Cos[1/7*Pi]*Cos[2/7*Pi]*Cos[3/7*Pi]];
> >
> > I try to simplify ff.
> >
> > In[194]:=
> > o1=FullSimplify[Together[TrigToExp[ff]]]
> > Out[194]=
> > (1/4)*Sqrt[1 + 1/Sqrt[2]]*Pi
> >
> > or as an another way take
> >
> > In[199]:=
> > o2=FullSimplify[TrigFactor //@ ff]
> > Out[199]=
> > (1/8)*Pi*Csc[Pi/8]
> >
> > o1 was obtained by FullSimplify[Together[TrigToExp[ff]]].
> >
> > Why doesn't
> >
> > In[206]:=
> > FullSimplify[ff, TransformationFunctions -> {Automatic, TrigToExp,
> > Together}]
> >
> > Out[206]=
> > Pi*Cos[Pi/7]*Cos[(2*Pi)/7]*Cos[(3*Pi)/7]*Csc[Pi*Cos[Pi/7]*Cos[(2*Pi)/
> > 7]*Cos[(3*Pi)/7]]
> >
> > do the same thing? What I miss here?
> >
> > Also, why the following does suceed?
> >
> > In[213]:=
> > FullSimplify[ff, TransformationFunctions -> {Automatic,
> > FullSimplify}]
> > Out[213]=
> > (1/8)*Pi*Csc[Pi/8]
> >
> > Thank you very much!
> >
> >



  • References:
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      • From: dimitris <dimmechan@yahoo.com>
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