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MathGroup Archive 2005

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Re: Re:Re: FullSimplify with Assumptions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg53447] Re: [mg53435] Re:Re: FullSimplify with Assumptions
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Wed, 12 Jan 2005 03:41:18 -0500 (EST)
  • References: <200501110631.BAA00751@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

On 11 Jan 2005, at 07:31, Goyder Dr HGD wrote:

> Thank you for your comments on my questions. However, this raises more 
> questions:
>
> 1. What is the mission of FullSimplify? I thought the mission was to 
> minimise the LeafCount of the expression. Are you are saying that this 
> is not possible if assumptions are included? If this is not possible 
> then what can FullSimplify hope to do?

FullSimplify tries to simplify an expression according to a user 
specified complexity function and using user specified transformation 
functions. Of course it uses certain defaults, which are selected 
because they have been judged most suitable for most users. You can 
inspect all the options of FullSimplify:


Options[FullSimplify]


{Assumptions :> $Assumptions, ComplexityFunction ->
    Automatic, ExcludedForms -> {},
   TimeConstraint -> Infinity, TransformationFunctions ->
    Automatic, Trig -> True}

As you can see the the default ComplexityFunction and 
TransformationFunctions are not specified (their value is Automatic) 
but Adam Strzebonski once posted the code of the default 
ComplexityFunction to this list. Any way, it is not actually LeafCount 
but is generally close to LeafCount. But of course the default values 
are not going to be suitable for most users. If they are not suited to 
your needs you have to define your own ComplexityFunction and your own 
TransformationFunctions. Moreover, anything that is included in the 
defaults will affect the vast majority of users who normally use only 
default values. It would obviously not be a good idea to choose 
defaults that might be ideal for one or two users and would cause other 
users programs to run much slower to no useful purpose.

Andrzej Kozlowski


>
> 2. If I change the input of FullSimplify from an expression to an 
> assertion to be tested, see below, then it seems to be able to use the 
> assumptions. Is this always going to be true?
>
> In[3]:=
> InputForm[FullSimplify[(L - L*y^2)/x^2, {-1 + x^2 + y^2 == 0}]]
>
> Out[3]//InputForm=
> (L - L*y^2)/x^2
>
> In[4]:=
> InputForm[FullSimplify[(L - L*y^2)/x^2 == L, {-1 + x^2 + y^2 == 0}]]
>
> Out[4]//InputForm=
> True
>

In this kind of cases (purely algebraic ones) it will always be true 
since in this case the answer is independent of the Groebner basis 
used.  What you are doing in essentially:



Last[PolynomialReduce[(L-L*y^2)/x^2-L,{-1+x^2+y^2},{x,y}]]


0


Last[PolynomialReduce[(L-L*y^2)/x^2-L,{-1+x^2+y^2},{y,x}]]


0

and the answer independent of the Groebner basis used even though, as I 
pointed out earlier:


Last[PolynomialReduce[(L - L*y^2)/x^2, {-1 + x^2 + y^2},
    {x, y}]]

L/x^2 - (L*y^2)/x^2


Last[PolynomialReduce[(L - L*y^2)/x^2, {-1 + x^2 + y^2},
    {y, x}]]

L

> 3. Why is it so impossible to check all lexical orderings? How would 
> the time increase with the number of variables and the number of 
> assumptions? Should this be an option, with warnings and time 
> constraints?

It is possible but you have to do it yourself. You can append suitable 
transformation functions to the default transformation functions.


>
> 4. The actual source of my problem was a huge trig function. I had 
> changed the trig functions into polynomials and the assumption above 
> was the remains of Sin[t]^2 + Cos[t]^2 == 1. FullSimplify did a lot of 
> good work before falling at the last hurdle.  How else can one 
> proceed?

The defaults are meant to be helpful to most people. They are not meant 
to solve all problems and clearly it would not be possible to to do 
that no matter what defaults where chosen. When they do not work in 
your particular situation you have to write your own ComplexityFunction 
and choose your own TransformationFunctions.


>
> Thanks
>
> Hugh Goyder
> -----Original Message-----
> From: Adam Strzebonski [mailto:adams at wolfram.com]
To: mathgroup at smc.vnet.net
> To: mathgroup at smc.vnet.net
> Subject: [mg53447] [mg53435] Re: [mg53339] FullSimplify with Assumptions
>
>
> Andrzej Kozlowski wrote:
>> *This message was transferred with a trial version of CommuniGate(tm) 
>> Pro*
>> On 7 Jan 2005, at 12:00, Goyder Dr HGD wrote:
>>
>>> In the examples below I would expect FullSimplify to give L.
>>> However, I get results that depend on the symbols I use.
>>> It would appear that symbols x and y are treated differently.
>>>
>>> How can I force FullSimplify to use the LeafCount as the
>>> ComplexityFunction?
>>>
>>>
>>> In[14]:= r1 = FullSimplify[(L - L*y^2)/x^2, {-1 + x^2 + y^2 == 0}]
>>>
>>> Out[14]= (L - L*y^2)/x^2
>>>
>>> In[15]:= r2 = FullSimplify[(L - L*x^2)/y^2, {-1 + x^2 + y^2 == 0}]
>>>
>>> Out[15]= L
>>>
>>> In[16]:= LeafCount[r1]
>>>
>>> Out[16]= 12
>>>
>>> In[17]:= LeafCount[r2]
>>>
>>> Out[17]= 1
>>>
>>> In[18]:= $Version
>>>
>>> Out[18]= "5.1 for Microsoft Windows (October 25, 2004)"
>>>
>>> Thanks for any comment
>>>
>>> Hugh Goyder
>>>
>> I would speculate that the reason has something to do with 
>> FullSimplify
>> using PolynomialReduce or related functions, whose outcome depends on
>> the ordering of the variables. Compare for example:
>>
>>
>> Last[PolynomialReduce[(L - L*y^2)/x^2, {-1 + x^2 + y^2},
>>    {x, y}]]
>>
>>
>> L/x^2 - (L*y^2)/x^2
>>
>> with
>>
>>
>> Last[PolynomialReduce[(L - L*y^2)/x^2, {-1 + x^2 + y^2},
>>    {y, x}]]
>>
>> L
>>
>> Note also that if you do not include explicitly the variables you 
>> will get:
>>
>>
>> Last[PolynomialReduce[(L - L*y^2)/x^2, {-1 + x^2 + y^2}]]
>>
>>
>> L/x^2 - (L*y^2)/x^2
>>
>>
>> I suspect this isn't a bug in FullSimplify. To remedy it FullSimplify
>> would need to use all possible orderings of variables when applying
>> algebraic functions whose output depends on variable order, and that 
>> is
>> obviously just not a reasonable option in terms of performance. Also,
>> Simplify and FullSimplify are not really optimized for conditions of 
>> the
>> form  something == something else; I find it a little surprising that 
>> it
>> works as well as it does. One should really deal with such problems by
>> using polynomial algebra, that is PolynomialReduce etc, with specified
>> variable ordering.
>>
>>
>> Andrzej Kozlowski
>> Chiba, Japan
>> http://www.akikoz.net/~andrzej/
>> http://www.mimuw.edu.pl/~akoz/
>>
>
> This is exactly the case.
>
> When FullSimplify is given equational assumptions it
> computes a single Groebner basis of the assumptions
> (with the DegreeReverseLexicographic monomial order
> and with the variables ordered by Sort).
> PolynomialReduce with respect to this Groebner basis
> is then applied as one of the transformation functions
> used to simplify subexpressions.
>
> Best Regards,
>
> Adam Strzebonski
> Wolfram Research
>
>
>
>
> -- 
> This message has been scanned for viruses and
> dangerous content by the Cranfield MailScanner, and is
> believed to be clean.
>


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