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MathGroup Archive 2006

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Re: How can I solve these simultaneous quadratic equations in Mathemetica?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65776] Re: How can I solve these simultaneous quadratic equations in Mathemetica?
  • From: bghiggins at ucdavis.edu
  • Date: Mon, 17 Apr 2006 02:28:20 -0400 (EDT)
  • References: <e1stfp$bf5$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Helsnam,

One way to at least get a sense of what the real roots look like is to
use FindRoot. We define the equations to be a function of c

eqnSet[c_] := {1 + 3c/2 + x(
    1 + c + y + z - x) + c*y + (y + z + x)/2 - w(2y + 2x - 3w + 3 + 2c)
==
           0,  c/2 + w(y + z - w + 3/2 + c) - x( 2w + 2z - 3x + 2) ==
0,
1 + 2c + w(1 + c) + z(w + x - z + 3/2 + c) -
       y(2w + 2z - 3y + 4 + 2c) == 0, y(x + w - y + 1 + c) - c(1 +
          w) + (y + w - x)/2 - z(2y + 2x - 3z) == 0};

Then use FindRoot to solve for the roots for a given value of C. Here
are the roots when C=2.8

sol1=FindRoot[eqnSet[28/10],{x,2},{y,2},{
      z,2},{w,2},WorkingPrecision->34,MaxIterations->10000]


{x -> -51.53066781230369784213550192724105,
y -> -29.11808168180036045675199024633007,
  z -> -26.97039854525144908482676830076015,
w -> -51.48887061457502546060305006442841}

Of course, the downside of using FindRoot is that different initial
guesses for the roots will lead to other solutions (if they exist).
Using NSolve gives all the roots. Here is a way to extract the real
roots from NSolve:

Cases[NSolve[eqnSet[2.8]], {Rule[_, _Real] ...}, 8]

{{z -> -94.1726, y -> -85.6825, x -> -171.641, w -> -167.993}, {
    z -> -26.9704, y -> -29.1181, x -> -51.5307, w -> -51.4889}, {
  z -> -4.55878, y -> -3.28643, x -> -5.09562, w -> -4.03022}}

Note for certain values of c there may be no real roots e.g. c=1. You
can also explore the behavior of a root by plotting it as a function of
c Here is how x varies with c in the range 0<c<0.99

ListPlot[Table[Flatten[{c, x /. Cases[NSolve[
      eqnSet[c]], {
          Rule[_, _Real] ...}, 8]}], {c, 0, 0.99, 0.05}], PlotJoined
-> True]


Hope this helps,

Brian


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