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Simultaneous equation problem - too many variables?

• To: mathgroup@smc.vnet.net
• Subject: [mg11331] Simultaneous equation problem - too many variables?
• From: "Richard Anderson" <richardj.anderson@stonebow.otago.ac.nz>
• Date: Fri, 6 Mar 1998 00:40:51 -0500

Greetings everyone,

I have a problem in population genetics that yields the following
equations:

Out[84]//InputForm=
us = (u*(1 + 2*v + h*t*(-1 + \[Theta])) +
    v*(1 + h*t*(-1 + \[Theta]) - t*\[Theta]))/
  (2 + u + v + t*(-2 + h*(u + v)*(-1 + \[Theta]) -
       v*\[Theta]))

Out[82]//InputForm=
vs = (u*(1 + 2*v + k*s*(-1 + \[Theta])) +
    v*(1 + k*s*(-1 + \[Theta]) - s*\[Theta]))/
  (2 + u + v + s*(-2 + k*(u + v)*(-1 + \[Theta]) -
       v*\[Theta]))

u and v are variables. k,h,s,t, and \[Theta] are constants (0 <= k <= 1,
0 <= h <= 1, -1 < s < 1, -1 < t < 1). I've tried using
SetAttributes[s,Constant] and so on, but that doesn't help with the
following problem.

I am interested in the solutions of the simultaneous equations {us == u,
vs == v}. (Please note that us and vs are separate variables, not u*s
and v*s). A previous question to MathGroup yielded the following idea
which works for simpler cases of the same kind of problem (the method
puts (us-u) and (vs -v) over a common denominator so that the problem
becomes one of solving for the numerator = 0).

equations ={us==u,vs==v};
polys=Map[#[[1]]-#[[2]]&,equations]; newpolys=
Map[Numerator[Together[#]]&,polys]; Reduce[newpolys==0,{u,v}];

However, in this case, Mathematica(v.3.0.1 running on a Pentium 200 MHz
and Win 95), runs out of memory, presumably because there are too many
variables. (As a concession, I can justifiably allow h=k, but if
possible would like to avoid it. Even with h=k, Mathematica fails to
solve) I've tried a number of substitutions - e.g. A = 1 + k s(-1 +
\[Theta]), B = A - t \[Theta] (and similarly in the equation for vs),
to no avail.

Any ideas? If anyone needs more information, please e-mail me at the
address below.

Thank you.

Cheers,

Richard

********************************************************************************
Richard Anderson
e-mail : richardj.anderson@stonebow.otago.ac.nz

• Follow-Ups:
  • Re: Simultaneous equation problem - too many variables?
    • From: Daniel Lichtblau <danl@wolfram.com>