Re: Re: Re: Simplifying in Mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg71257] Re: [mg71247] Re: [mg71200] Re: Simplifying in Mathematica
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sun, 12 Nov 2006 06:47:49 -0500 (EST)
• References: <eish5b\$nq1\$1@smc.vnet.net> <200611101138.GAA13711@smc.vnet.net> <200611110839.DAA27617@smc.vnet.net> <089AA94F-FCDE-44A2-9239-A5C30CDBCAD1@mimuw.edu.pl>

```On 11 Nov 2006, at 21:56, Andrzej Kozlowski wrote:

>
> On 11 Nov 2006, at 17:39, Daniel Lichtblau wrote:
>
>> pierodancona at gmail.com wrote:
>>> Define a function (depending on the variables B,b,A,a,c,d...etc,
>>> all of them) equal to your expression, and compute it at many
>>> points.
>>> If the expression is 0, the result should always be 0. Notice
>>> that if
>>> your expression is a polynomial or rational and you choose your
>>> points in a "not too special" way, this gives actually a rigorous
>>> "proof"
>>> that the expression is 0. In general, this is an extrremely reliable
>>> heuristic (and of course a "proof" if you do not get 0 for some
>>> values).
>>>
>>> I encounter the same problem frequently, and this method works
>>> very well, also to test if two complicated expressions are the same
>>> or not (guess how :)
>>>
>>> Piero
>>>
>>>
>>>
>>> 330006 at gmail.com wrote:
>>>
>>>> I have a function which is a sum of many terms which look like
>>>> this:
>>>>
>>>> (2*(B-b)^2 - 2*(A-a)*c*d^2)/(4*b^2*(1-c*2)*d^2)
>>>>
>>>> I think the function is actually equal to 0, but I have a hard
>>>> time in
>>>> trying to simplify it in Mathematica. Any ideas or commands I
>>>> should
>>>> try? Any suggestions in general about simplifying formulas will
>>>> also
>>>> be greatly appreciated!
>>>>
>>>> Thanks a lot!
>>
>>
>> Random point evaluation, and various other heuristics, are what lie
>> behind the Mathematica predicate Developer`ZeroQ.
>>
>>
>> Daniel Lichtblau
>> Wolfram Research
>>
>
> Another possible approach is to evaluate NMaximize[Abs[expr],vars],
> where expr is the expression suspected of being zero and vars the
> variables.
>
> Andrzej Kozlowski

I should have added, however, that in such cases one should not
resort to numerical methods unless and until one has really exhausted
all the symbolic ones. There are many things one could try to show
that an algebraic expression is zero and unless it involves some
special functions etc. numerical methods should not be needed. But of
course it is impossible to give any sensible advice beyond this
without knowing more about the expression in question.

Andrzej Kozlowski
Tokyo, Japan

```

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