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**:**questions***From:*dimitris <dimmechan@yahoo.com>