Re: Re: Strange behaviour of Simplify
- To: mathgroup at smc.vnet.net
- Subject: [mg79117] Re: [mg79086] Re: Strange behaviour of Simplify
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Thu, 19 Jul 2007 03:26:51 -0400 (EDT)
- References: <200707150514.BAA08789@smc.vnet.net><f7f25j$muj$1@smc.vnet.net> <200707180656.CAA04448@smc.vnet.net>
On 18 Jul 2007, at 07:56, Andreas Maier wrote:
> On Jul 16, 8:10 am, Andrzej Kozlowski <a... at mimuw.edu.pl> wrote:
>> On 15 Jul 2007, at 14:14, Andreas Maier wrote:
>>
>>
>>
>>> Hi,
>>
>>> i'm trying to simplify the following expression
>>
>>> In:= gamma=c Sqrt[g00])/Sqrt[c^2 g00 + g11 v^2]
>>> In:=LeafCount[gamma]
>>> Out:=52
>>
>>> using Simplify
>>
>>> In:= gamma2=Simplify[gamma, {Sqrt[g00] > 0, c > 0 }]
>>
>>> Out:=c/Sqrt[c^2 + (g11 v^2)/g00]
>>> In:=LeafCount[gamma2]
>>> Out:=44
>>
>>> Why is Mathematica (I'm using V6.0) able to cancel Sqrt[g00], but is
>>> not
>>> able to cancel c? I also tried ComplexityFunction->LeafCount, which
>>> should work, because
>>
>>> In:=gamma3=1/Sqrt[1 + (g11 v^2)/(c^2 g00)]
>>> In:=LeafCount[gamma3]
>>> Out:=43
>>
>>> the LeafCount of gamma3 is smaller than gamma2, but it doesn' work.
>>> Can anybody tell me, how i can transform gamma to gamma3 in
>>> Mathematica?
>>
>>> Andreas Maier
>>
>> There is nothing strange about this behaviour. Simplify only uses
>> certain specified transformation functions and applies certain
>> sequences of them (such that the complexity does not increase at any
>> step) but there is no such sequence of functions that would do what
>> you want, because any such sequence would have to temporarily inrease
>> the complexity of the expression. Rather than spend time and energy
>> on trying to find a suitable mix of TransformationFunctions and a
>> ComplexityFunction that would produce the result you want, it is much
>> simpler to use replacement rules or a combination of replacement
>> rules and Simplify, like this:
>>
>> Simplify[gamma /. c -> Sqrt[t], {Sqrt[g00] > 0, t > 0}] /. t -> c^2
>>
>> 1/Sqrt[(g11*v^2)/(c^2*g00) + 1]
>>
>> Andrzej Kozlowski
>
> Actually i figured that out by myself just before i read your message.
> But thank you anyway.
> Still i think this solution is not a very elegant one, since it
> involves a lot of guess work
> (Although in this case it was suggestive to try to substitute with a
> Sqrt[], but in general this might not be the case).
> Is there no possibility in Mathematica to manually move out certain
> factors from the enumerator and
> denominator and then let Simplify cancel these factors? I understand
> that Simplify cannot do transformations
> that temporarily generate a higher LeafCount, but it would be nice, if
> i could generate these expressions manually
> to give a new starting point for Simplify, so it can find expressions
> with an even lower LeafCount.
>
> Andreas Maier
>
>
>
As I have argued many times here, the purpose of Simplify is not to
transform an expression into some form that is already known to the
user. The right tool for this sort of thing are pattern matching and
transformation rules. You can sometimes makes things a little simpler
(at least as far as the amnount of typing is concerned) by a
judicious use of Simplify but as you correctly note this involves
guess work, since it is hard to tell exactly what output Simplify
will return (because it is intended for quite a different purpose).
So, in my opinion, the best way to do this sort of thing is as you
would do it "by hand", and in this case I would use:
Numerator[gamma]/(c*Sqrt[g00])/
Sqrt[Apart[Denominator[gamma][[1]]/(c^2*g00)]]
1/Sqrt[(g11*v^2)/(c^2*g00) + 1]
You can always use Simplify to verify that what you got is equal to
gamma:
Simplify[% == gamma, { c > 0, Sqrt[g00] > 0}]
True
Andrzej Kozlowski
- References:
- Strange behaviour of Simplify
- From: Andreas Maier <andimai@web.de>
- Re: Strange behaviour of Simplify
- From: Andreas Maier <andimai@web.de>
- Strange behaviour of Simplify