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MathGroup Archive 2005

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Re: silly questions?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg59115] Re: silly questions?
  • From: snoofly <snoofly at snoofly.com>
  • Date: Fri, 29 Jul 2005 00:41:53 -0400 (EDT)
  • References: <200507270526.BAA20063@smc.vnet.net> <dc9vhu$cnk$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Thanks Andrzej and others for your informative replies on LeafCount  
regarding my simplification query. I see how it all makes sense regarding  
the Cos simplification but I have one little nagging issue left as the  
following example shows:

Factor[x^4 - 16]

(-2 + x)*(2 + x)*(4 + x^2)

FullSimplify[(x^4-16)/(-2+x)]

(2 + x)*(4 + x^2)

FullSimplify[(x^4-16)/(2+x)]

(-16 + x^4)/(2 + x)

LeafCount[(-16+x^4)/(2+x)]

11

LeafCount[(-16+x^4)/(-2+x)]

11

LeafCount[(2+x)*(4+x^2)]

9

LeafCount[(-2+x)*(4+x^2)]

9

The question to you gifted individuals is why doesn't  
FullSimplify[(x^4-16)/(2+x)] give (-2 + x)*(4 + x^2) which has a lower  
leaf count?

On Thu, 28 Jul 2005 07:55:58 +0100, Andrzej Kozlowski <akoz at mimuw.edu.pl>  
wrote:

>
>
> On 27 Jul 2005, at 07:26, Kent Holing wrote:
>
>> Why does not (x^5-32)/(x-2)//FullSimplify in Mathematica  work?
>> Compare with Factor[x^5-32]//InputForm which returns (-2 + x)*(16 +
>> 8*x + 4*x^2 + 2*x^3 + x^4).
>> So why does not the first command just return 16 + 8*x + 4*x^2 +
>> 2*x^3 + x^4?
>> As in a factorization above, how is the easiest way to pick
>> automatically (by a function) the factors of say degree >=2,  if any ?
>>
>> Kent Holing
>>
>>
>
> Very "simple". What makes you think the cancelled out form is "simpler"?
>
>
> LeafCount[(x^5-32)/(x-2)]
>
>
> 11
>
> while
>
>
> LeafCount[Cancel[(x^5-32)/(x-2)]]
>
>
> 18
>
> The cancelled form is much more "complicated", at least as measured
> by LeafCount (and Mathematica's default complexity function).
>
> So if you want your answer it is better to make ask Mathematica to
> make the expression more "complex":
>
>
> Simplify[(x^5 - 32)/(x - 2), ComplexityFunction ->
>     (1/LeafCount[#1] & )]
>
>
> x^4 + 2*x^3 + 4*x^2 + 8*x + 16
>
> But it is of course much more sensible to just use Cancel.
>
> Andrzej Kozlowski
>
>
>


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