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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: [mg65902] Re: How can I solve these simultaneous quadratic equations in Mathemetica?
  • From: "Chris Chiasson" <chris.chiasson at gmail.com>
  • Date: Sat, 22 Apr 2006 04:54:22 -0400 (EDT)
  • References: <200604160748.DAA11097@smc.vnet.net><e22hah$e5h$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Here is a way to get a function/plot of the variables vs c by combining
Lichtblau's NSolve with Woll's homotopic continuation:

initc is the point where NSolve will be called to give an initial
condition
halfinterval is the range of c to the right and left of initc

eqns[4] is keyed to find a Real solution for the initial condition

eqns[1]={1+3c/2+x(1+c+y+z-x)+c*y+(y+z+x)/2-w(2y+2x-3w+3+2c)\[Equal]0,
      c/2+w(y+z-w+3/2+c)-x(2w+2z-3x+2)\[Equal]0,
      1+2c+w(1+c)+z(w+x-z+3/2+c)-y(2w+2z-3y+4+2c)\[Equal]0,
      y(x+w-y+1+c)-c(1+w)+(y+w-x)/2-z(2y+2x-3z)\[Equal]0};
reps[1]=Cases[Variables[eqns[1]/.Equal\[Rule]List],
      blah_Symbol\[RuleDelayed]Rule[blah,blah[c]]/;blah=!=c];
eqns[2]=eqns[1]/.reps[1];
eqns[3]=D[#,c]&/@eqns[2];
solvars[1]=Last/@reps[1];
initc=1/2;
halfinterval=3/8;
crangespec={c,initc-halfinterval,initc+halfinterval};
eqns[4]=Equal@@@
    First[Cases[Apply[NSolve,{eqns[2],solvars[1]}/.c\[Rule]initc],
        Map[Rule[#,_Real]&,solvars[1]/.c->initc]]]
solns[1]=NDSolve[eqns/@Range[3,4]/.List\[Rule]And,Head/@solvars[1],
      Evaluate[crangespec]];
Plot[Evaluate[solvars[1]/.solns[1][[1]]],Evaluate[crangespec],
    PlotRange\[Rule]{0,50},
    PlotStyle\[Rule](Hue[N@#/Length[solvars[1]]]&/@Range[4]),
    ImageSize\[Rule]400];


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