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MathGroup Archive 2005

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Re: problem solving polynomial equations


On 15 Oct 2005, at 11:22, wtplasar at ehu.es wrote:

>
> Hi,
>
>
> I have  two equations. The first one is
>
> eqy=(y*(3*b^2 + 12*b*y^2 + 12*y^4 + 3*b^2*z^4 + 4*b*y^2*z^4 +
> 3*r*y*Sqrt[4*y^4 + 4*b*y^2*(1 + z^4) + b^2*(1 + 3*z^4) + z^2*(8*b*y^2
> + 4*y^4 + b^2*(3 + z^4))*Sign[k]]*
>     Sign[s] + z^2*Sign[k]*(6*b^2 + 16*b*y^2 + 8*y^4 + r*y*Sqrt[4*y^4 +
> 4*b*y^2*(1 + z^4) + b^2*(1 + 3*z^4) + z^2*(8*b*y^2 + 4*y^4 + b^2*(3 +
> z^4))*Sign[k]]*
>       Sign[s])))/(3*(3*b^2 + 8*b*y^2 + 4*y^4 + 3*b^2*z^4 + 2*b*(3*b +
> 4*y^2)*z^2*Sign[k] +
>    2*r*y*Sqrt[4*y^4 + 4*b*y^2*(1 + z^4) + b^2*(1 + 3*z^4) + z^2*
> (8*b*y^2 + 4*y^4 + b^2*(3 + z^4))*Sign[k]]*Sign[s]));
>
>
> and the second one is
>
> eqz=(y*z*(4*b*y + 8*y^3 + 4*b*y*z^4 + r*Sqrt[4*y^4 + 4*b*y^2*(1 + z^4)
> + b^2*(1 + 3*z^4) + z^2*(8*b*y^2 + 4*y^4 + b^2*(3 + z^4))*Sign[k]] 
> *Sign
> [s] +
>    z^2*Sign[k]*(8*b*y + 8*y^3 + r*Sqrt[4*y^4 + 4*b*y^2*(1 + z^4) +  
> b^2*
> (1 + 3*z^4) + z^2*(8*b*y^2 + 4*y^4 + b^2*(3 + z^4))*Sign[k]]*Sign
> [s])))/
>  (3*(3*b^2 + 8*b*y^2 + 4*y^4 + 3*b^2*z^4 + 2*b*(3*b + 4*y^2)*z^2*Sign
> [k] +
>    2*r*y*Sqrt[4*y^4 + 4*b*y^2*(1 + z^4) + b^2*(1 + 3*z^4) + z^2*
> (8*b*y^2 + 4*y^4 + b^2*(3 + z^4))*Sign[k]]*Sign[s]));
>
> Now when I do
>
> Solve[{eqy == 0, eqzsubs == 0}, {y, z}] /. Sign[s]^2 -> 1
>
> I get the result fairly quickly, but if I do
>
> Solve[{Numerator[eqy]== 0, Numerator[eqz] == 0}, {y, z}]/. Sign[s]^2 -
>
>> 1
>>
>
> it seems to get stuck. I may get an answer eventually, but it seemed
> to be taking too long and aborted it.
>
> Any clues? Thanks.
>
> Ruth
>
>

Basically what happens is this. Solve can't find the general solution  
to parametric equations of this kind. (It is not the fault of Solve;  
there is really no way to do it).  So what it does is returns  
basically any solutions it can find, by using certain "heuristic"  
methods. One method is simply to set some variables to 0 and see if  
the equations can be solved for the remaining ones. This is what  
seems to happen here:


Solve[{eqy == 0, eqz == 0}, {y, z}] /. Sign[s]^2 -> 1


{{y -> (1/4)*((-r)*Sign[s] - Sqrt[r^2 - 8*b]), z -> 0},
   {y -> (1/4)*(r*Sign[s] - Sqrt[r^2 - 8*b]), z -> 0},
   {y -> (1/4)*(Sqrt[r^2 - 8*b] - r*Sign[s]), z -> 0},
   {y -> (1/4)*(r*Sign[s] + Sqrt[r^2 - 8*b]), z -> 0},
   {y -> 0}}

These are the kind of solutions you could find yourself by hand!

Now for the other case.

Solve[{Numerator[eqy] == 0, Numerator[eqz] == 0}, {y, z}] /. Sign[s] 
^2 -> 1

This is pure speculation, but I would guess Solve has for some reason  
started to apply some algorithm that might give additional solutions.  
Unfortunately there is no reason to believe that the computation will  
finish in reasonable time. This could in fact be one of the cases  
where "better is worse", meaning that the fact that the equations are  
"better" (so Mathematica makes a serious attempt to find solutions)  
cases no solutions to be returned. To see this phenomenon in an  
extreme form on a trivial example, compare these two equations:



Solve[x (Exp[Sin[x]-Cos[x]])\[Equal]0,x]

Inverse functions are
     being used
       by Solve, so some solutions may not be \
found; use Reduce for complete solution information.

{{x->0}}

Solve can do nothing with this except find the trivial solution so it  
quickly returns the trivial solution. Now consider the following  
equation that  I made up at random:

Solve[(x^32 - x^31)^(1/12) + (x^7 - x^11/3)^(1/5) + 17x + 5 == 0, x]

This is "better" than the other one, since Mathematica, in principle  
knows algorithms for solving equations of this type. But it is  
"worse" since the computation is unlikely to finish (I did not wait  
long  but even if this ever finishes one can produce a more  
complicated example that will not).

So, I think what happened in your case is that your second set of  
equations is sufficiently "better" than the first to make it actually  
"worse".

Well, of course, this is only speculation. Daniel Lichtblau can  
almost certainly  tell if it is true, if he chooses to.

Andrzej Kozlowski










Andrzej Kozlowski
Tokyo, Japan




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