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Re: Simplify
*To*: mathgroup at smc.vnet.net
*Subject*: [mg98677] Re: [mg98659] Simplify
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Thu, 16 Apr 2009 04:13:23 -0400 (EDT)
*References*: <200904150902.FAA08151@smc.vnet.net> <ACCF1343-E098-469B-A599-4CA4F9E1A5E0@mimuw.edu.pl> <49E5CA92.7030005@metrohm.com>
Yes, you are right but I still think it is beside the point. Consider
this rather more striking (in my opinion) example.
let
expr = 2^(z + 1)*3^z
You can easily see that this can be re-written as 2*6^z and that:
LeafCount[Hold[2*6^z]]
6
while
LeafCount[expr]
9
so 2*6^z has a much smaller LeafCount than expr, but you can never
Simplify expr to 2*6^z because the Evaluator will always rewrite it
again as expr. This has a rather unpleasant consequence:
FullSimplify[2*6^z - 6^z]
2^(z + 1)*3^z - 6^z
Simplify cannot see that the simplest answer if 6^z (no matter what
ComplexityFunction you use) because immediately on evaluation 2*6^z is
converted to expr above and then it is too late; Simplify can't see
that the first term is simply twice the second term. Yet Simplify has
no problem noticing that:
Simplify[2*6^z - 6^z == 6^z]
True
The reason for the problem is the following "canonical form":
expressions like (a^k1*b^k2*c^k3...)^z *(a*l1*b^l2*...) where a, b, c
are positive are re-written as a^(k1+l1)*b^(k2+l2)*....
For example
(2^u*5^v)*(2*3^w*5^z)
2^(u + 1) 3^w 5^(v + z)
(actually the rule used is note general, but I just want to give an
example of a "canonical form" that causes trouble).
Since these reductions are made by the Evaluator, Simplify has no
effect on them. But because of them we get the following inconsistent
behaviour:
2*5^z - 5^z
5^z
but
2*6^z - 6^z
2^(z + 1)*3^z - 6^z
This, of course, cannot be changed by applying Simplify with any
ComplexityFunction because canonical forms can't be changed by
Simplify (unless you apply Hold).
Andrzej
On 15 Apr 2009, at 20:52, dh wrote:
> Hi Andrzej,
> thanks for your helpfull explanations.
> Let me just mention that there is an additional reason, internally
> the expression x^2/y^2 is rewritten and has the the same leaf count
> as (x/y)^2. Consider:
> FullForm[Hold[x^2/y^2]]
> FullForm[x^2/y^2]
> Daniel
>
>
> Andrzej Kozlowski wrote:
>> Actually, this point has been explained many times (by me ;-))
>> (I like to think of Mathematica's evaluation process in terms of
>> something called "The Evaluator", which I think I first found in
>> David Wagner's book "Power Programming with Mathematica". I think
>> it is only an abstraction, along with "the Parser", "the
>> Typesetter" etc, but a convenient one when one is thinking about
>> the evaluation process. )
>> The "Evaluator" always evaluates
>> (x/y)^2
>> to
>> x^2/y^2
>> This happens before Simplify takes any effect. Even if Simplify
>> converted x^2/y^2 to (x/y)^2 the Evaluator would kick in and again
>> convert it back to x^2/y^2. Since the Evaluator always overrides
>> Simplify there is no way to get (x/y)^2 as the output without using
>> Hold.
>> Perhaps you are asking why Mathematica (or the "Evaluator")
>> automatically converts (x/y)^2 to x^2/y^2. It's because of
>> something called "canonical forms" or "standard forms". Basically,
>> in order to optimize performance in computer algebra systems one
>> want to reduce to 0 as quickly as possible as many expressions that
>> are actually equal. The earlier you do this the better the
>> performance. If you allow expressions to contain a large number of
>> subexpressions that are actually 0 until you apply Simplify, it may
>> very seriously impair performance. "Canonical forms" (or "normal
>> forms", there is a slight difference between them but I shall
>> ignore it) are certain unique forms to which various expressions
>> are reduced automatically by the Evaluator (before applying
>> Simplify). This has the effect causing cancellations to occur
>> early. Also, these "canonical forms" have to be independent of any
>> particular ComplexityFunction used, hence the reduction has to be
>> performed outside Simplify. The advantage of using canonical forms
>> independent of ComplexityFunction is that they often enable
>> Mathematica to identify two expressions as equal even if Simplify
>> can't fine a sequence of Complexity reducing transformations that
>> will convert one expression into the other.
>> Not surprisingly, using "canonical forms" can sometimes produce
>> undesirable side-effects and this is one of them (a rather minor
>> one, worse ones do occur).
>> Andrzej Kozlowski
>> Andrzej Kozlowski
>> On 15 Apr 2009, at 18:02, dh wrote:
>>>
>>>
>>> Hi,
>>>
>>> can somebody explain, why
>>>
>>> Simplify[x^2/y^2,ComplexityFunction->LeafCount]
>>>
>>> does not simplify to (x/y)^2, although the LeafCount is:
>>>
>>> LeafCount[Hold[x^2/y^2]] gives 10
>>>
>>> and
>>>
>>> LeafCount[Hold[(x/y)^2]] gives 8
>>>
>>>
>>>
>>> Daniel
>>>
>>>
>>>
>
>
> --
>
> Daniel Huber
> Metrohm Ltd.
> Oberdorfstr. 68
> CH-9100 Herisau
> Tel. +41 71 353 8585, Fax +41 71 353 8907
> E-Mail:<mailto:dh at metrohm.com>
> Internet:<http://www.metrohm.com>
>
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