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RE: RE: [Q] Equation solving?

• To: mathgroup at smc.vnet.net
• Subject: [mg23359] RE: [mg23318] RE: [mg23282] [Q] Equation solving?
• From: "David Park" <djmp at earthlink.net>
• Date: Thu, 4 May 2000 02:59:26 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

> From: Andrzej Kozlowski [mailto:andrzej at tuins.ac.jp]
To: mathgroup at smc.vnet.net

> I wrote the message below in a hurry and did not express it very
> well. Here
> is another go. We can get the answer slightly faster if we do the
> following:
>
> In[1]:=
> eqns = {Q1 == (2*z + 1 - 1)^2 + 4*1*1,
>     k1 == 1/Sqrt[Q1]*(v*z - 3/2*(2*z + 1 + 1 -
>          Sqrt[Q1])), Q2 == (2*z + 1 - 1)^2 + 4*1*1,
>     k2 == 1/Sqrt[Q2]*((6.1 - v)*z -
>        2/2*(2*z + 1 + 1 - Sqrt[Q2])),
>     3/2*(2*z + 1 + 1) + (k1 - 3/2)*Sqrt[Q1] ==
>      6.1*z - (2/2*(2*z + 1 + 1) + (k2 - 2/2)*Sqrt[Q2])};
> In[2]:=
> sols = Solve[eqns, {k1, k2, z}, {Q1, Q2, w}]
> Out[2]=
> {{k1 -> 0., k2 -> 0., z -> 0.}, {k1 -> 0., k2 -> 2., z -> 0.}, {k1 -> 3.,
>     k2 -> 0., z -> 0.}, {k1 -> 3., k2 -> 2., z -> 0.}}
>
> However, only the first solution is genuine, the rest are spurious (or
> parasites).

Thanks Andrzej,

We should all learn to check solutions. This would have been a smoother way
to the one genuine solution. Eliminate k1 and k2 first and solve for Q1, Q2
and z. Then solve for k1 and k2.

Eliminate[eqns, {k1, k2}]
sols = Solve[%, {Q1, Q2, z}]

Q1 == 4. && Q2 == 4. && z == 0. && Q1 != 0. && Q2 != 0.
{{Q1 -> 4., Q2 -> 4., z -> 0.}}

Rationalize[Chop[eqns /. sols]]
sols2 = Solve[%[[1]]]

{{True, k1 == 0, True, k2 == 0, 3 + 2*(-(3/2) + k1) ==
    -2 - 2*(-1 + k2)}}
{{k1 -> 0, k2 -> 0}}

Now check the solution.

eqns /. sols /. sols2 // Chop
{{{True, True, True, True, True}}}

I guess that the lesson is: if you are having difficultly solving a set of
equations, try doing it in stages, but always check the solutions, which I
failed to do.

David Park
djmp at earthlink.net
http://home.earthlink.net/~djmp/