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Re: Re: Eliminating common factors?

• To: mathgroup at smc.vnet.net
• Subject: [mg93361] Re: [mg93354] Re: Eliminating common factors?
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Wed, 5 Nov 2008 04:52:31 -0500 (EST)
• References: <200811041121.GAA29397@smc.vnet.net>

On 4 Nov 2008, at 20:21, AES wrote:

> I appreciate the effort Bob Hanlon put into listing multiple solutions
> to my original post in his response reproduced below.  My actual
> calculation, which I simplified for the initial post, was
>
>   sx0sqr = wx0sqr/4; sy0sqr = wy0sqr/4;
>   sxsqr = sx0sqr + (mx4 lamsqr (z - z0x)^2/(16 pisqr sx0sqr));
>   sysqr = sy0sqr + (my4 lamsqr (z - z0y)^2/(16 pisqr sy0sqr));
>   srsqr = sxsqr + sysqr;
>   sr0sqr = sx0sqr + sy0sqr;
>   wxsqr = 4 sxsqr; wysqr = 4 sysqr;
>  {wxsqr, wysqr}
>
> leading to the output
>
>   {4 (wx0sqr/4 + (lamsqr mx4 (z - z0x)^2)/(4 pisqr wx0sqr)),
>     4 (wy0sqr/4 + (lamsqr my4 (z - z0y)^2)/(4 pisqr wy0sqr))}
>
> This is just the first cell of a lengthy and messy symbolic  
> calculation
> of optical beam propagation formulas, in which the objective is to
> transform these results in x,y coordinates into comparable formulas in
> cylindrical coordinates and derive relations between rectangular and
> cylindrical beam parameters.
>
> By accident as much as design, the outputs above, except for the
> superfluous factors of 4, come out in just the form conventionally  
> used
> in the relevant literature, namely a "constant" or z-independent term,
> plus a separate z-dependent term containing the physically significant
> factors (z-z0x)^2 and (z-z0y)^2, which I'd like to preserve intact  
> since
> they'll provide helpful physical insight in messier results later on.
>
> The problem is that _every_one_ of the commands suggested below (plus
> the FactorTerms command suggested by someone else), while getting  
> rid of
> the unnecessary factors of 4, also expands out the (z-z0)^2
> factors into 3 separate terms, thus making the results longer,  
> messier,
> and less easy to view.
>
>
> One solution would be to replace each of the (z-z0)^2 factors by a
> single factor zminusz0sqr in my initial inputs, then convert back at  
> the
> very end of the calculations.  Doing this will make any of the Hanlon
> suggestions work as desired.

Actually, there is no need to do that.



Simplify[{4*(wx0sqr/4 + (lamsqr*mx4*(z - z0x)^2)/
             (4*pisqr*wx0sqr)),
      4*(wy0sqr/4 + (lamsqr*my4*(z - z0y)^2)/
             (4*pisqr*wy0sqr))}, ExcludedForms -> {_Symbol}]

{(pisqr*wx0sqr^2 + lamsqr*mx4*(z - z0x)^2)/
      (pisqr*wx0sqr), (pisqr*wy0sqr^2 +
         lamsqr*my4*(z - z0y)^2)/(pisqr*wy0sqr)}


Andrzej Kozlowski


>
>
> I realize it's pointless to ask why Mathematica doesn't automatically
> remove the utterly superfluous integer factors of 4 in the above
> outputs.  In fact, I still wonder if there is some deep, deep reason
> that I don't grasp that says that Mathematica _should_not_ do this
> apparently obvious simplification, because doing it could possibly  
> lead
> to trouble in some other situation if Mathematica were designed to do
> this automatically.
>
>
>
> =============================================
>> From Bob Hanlon:
>
> In the simple form provided, just about any command that touches it  
> will
> work. Without the actual expression there is no way to help.
>
> expr = 4 (a/4 + b/(4 c))
>
> 4*(a/4 + b/(4*c))
>
> expr // Simplify
>
> a + b/c
>
> expr // Apart
>
> a + b/c
>
> expr // Expand
>
> a + b/c
>
> expr // ExpandAll
>
> a + b/c
>
> expr // Cancel // Apart
>
> a + b/c
>
> expr // Together // Apart
>
> a + b/c
>
> expr // Factor // Apart
>
> a + b/c
>
> Bob Hanlon
>

• References:
  • Re: Eliminating common factors?
    • From: AES <siegman@stanford.edu>