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MathGroup Archive 2001

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Re: Solutions that are not solutions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg32086] Re: [mg32049] Solutions that are not solutions
  • From: "PSi" <no at tee.gr>
  • Date: Sat, 22 Dec 2001 04:23:06 -0500 (EST)
  • References: <B44315B6-F667-11D5-9161-00039311C1CC@tuins.ac.jp>
  • Sender: owner-wri-mathgroup at wolfram.com

Many thanks!
The VerifySolutions -> True should probably be necessary for any
application of Solve.
However I can't find it in the online help!

----- Original Message -----
From: "Andrzej Kozlowski" <andrzej at tuins.ac.jp>
To: mathgroup at smc.vnet.net
Subject: [mg32086] Re: [mg32049] Solutions that are not solutions


> You can see more clearly what is going on by using Reduce instead of
> Solve:
>
> In[1]:=
> X = {{0, y, z}, {y, x, t}, {u, v, w}};
> A = {{1, 1, a}, {0, 1, 0}, {0, 0, 1}};
> Reduce[{X . A == Transpose[A] . X, Det[X] == 1},
>    {x, y, z, t, u, v, w}, VerifySolutions -> True]
>
> Out[3]=
> a == 0 && u == 0 && w == -(1/y^2) && z == 0 ||
>    u == a*y && x == (-1 + a*t*y^2 + a*v*y^2 - w*y^2)/
>       (a^2*y^2) && z == a*y && a != 0 && y != 0
>
> You see that the first solution works only if a=0. So the "spurious"
> solution that you get actually is a solution that works only in the
> special case a==0. In general Solve tries to avoid such non-generic
> solutions. However, in some situations it is possible that certain
> variables get eliminated during the process of finding a solution
and
> then one may obtain non-generic solutions with respect to these
> eliminated variables (parameters). Note that in your equation the
number
> of solve variables is larger than the number independent variables.
You
> can avoid getting this non-generic  solution  by reducing the number
of
> solve variables in your equations to just three and treating all the
> others as parameters, e.g. :
>
> In[35]:=
> Solve[{X . A == Transpose[A] . X, Det[X] == 1}, {x, y, z},
>    VerifySolutions -> True]
>
> Out[35]=
> {{x -> (-a^2 + a*t*u^2 + a*u^2*v - u^2*w)/(a^2*u^2), z -> u,
>     y -> u/a}}
>
> In[36]:=
> Simplify[{X . A == Transpose[A] . X, Det[X] == 1} /. %]
>
> Out[36]=
> {{True, True}}
>
> Andrzej Kozlowski
> Toyama International University
> JAPAN
> http://platon.c.u-tokyo.ac.jp/andrzej/




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