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MathGroup Archive 1998

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Re: Solving simultaneous eqns

  • To: mathgroup at smc.vnet.net
  • Subject: [mg14285] Re: [mg14199] Solving simultaneous eqns
  • From: Jurgen Tischer <jtischer at col2.telecom.com.co>
  • Date: Mon, 12 Oct 1998 13:52:02 -0400
  • Sender: owner-wri-mathgroup at wolfram.com

Hi Yun,
I have tried to solve your equations, but my solution is not very
convincing. I add the notebook anyway.

Jurgen

Yeoung-Sang Yun wrote:

> Hello!
>
> I want to know how to solve the following simulaneous equations in
> Mathematica:
>
> x=x^1.2/(2*x^3+y^0.7+4*z^2.5)
> y=y^0.7/(2*x^0.6+y^2+z^2.2)
> z=0.9*z^1.5/(x^0.7+2*y^0.2+z^1.1)
>
> Thanks for helping.
> Y.-S. Yun
> Department of Chemical Engineering
> Pohang University of Science and Technology San 31, Hyoja-dong, Pohang
> 790-784, Republic of Korea


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Notebook[{
Cell["\<\
To facilitate writing for me I changed the equations to a function whose
\ zeros are the solutions of your problem. \ \>", "Text"],

Cell[BoxData[
    \(f[x_, y_, z_] = {x - x\^1.2\/\(2\ x\^3 + y\^0.7 + 4\ z\^2.5\), 
        y - y\^0.7\/\(2\ x\^0.6 + y\^2 + z\^2.2\), 
        z - \(0.9\ z\^1.5\)\/\(x\^0.7 + 2\ y\^0.2 + z\^1.1\)}\)],
"Input"],

Cell["\<\
First I observe that for every two variables zero I can calculate a
solution.\ \
\>", "Text"],

Cell[BoxData[
    \(Solve[f[x, 0, 0] == 0]\)], "Input"],

Cell[BoxData[
    \(Solve[f[0, y, 0] == 0]\)], "Input"],

Cell[BoxData[
    \(Solve[f[0, 0, z] == 0]\)], "Input"],

Cell["\<\
Now I'm just searching around with FindRoot. (I checked with much more \
values, this is just the result.)\
\>", "Text"],

Cell[BoxData[
    \(FindRoot[f[x, y, z], {x, 1}, {y, 1}, {z, 0}, MaxIterations -> 100]
// 
      Chop\)], "Input"],

Cell[BoxData[
    \(FindRoot[f[x, y, z], {x, 5}, {y, 6}, {z, 1}, MaxIterations -> 100]
// 
      Chop\)], "Input"],

Cell[TextData[{
  " I start having the idea there are no other solutions with
z\[NotEqual]0 \ than the one already found. Now if  ",
  Cell[BoxData[
      \(TraditionalForm\`z == 0\)]],
  ", the third equation is satisfied, so to look for posible solutions I
look \ at the intersections of the zeros of the first and the second
equation and \ find there are two solutions, the two I found already."
}], "Text"],

Cell[BoxData[
    \(<< Graphics`\)], "Input"],

Cell[BoxData[
    \(ImplicitPlot[{Evaluate[\(f[x, y, 0]\)[\([1]\)]] == 0, 
        Evaluate[\(f[x, y, 0]\)[\([2]\)]] == 0}, {x, 0.0000001, 2}, {y, 
        0.00001, 2}]\)], "Input"],

Cell[TextData[{
  "To see what happens for ",
  Cell[BoxData[
      \(TraditionalForm\`z > 0\)]],
  " I use ContourPlot3D whose default is to show the 0-surface." }],
"Text"],

Cell[BoxData[
    \(ContourPlot3D[
      Evaluate[\(f[x, y, z]\)[\([1]\)]], {x,  .0000001, 1.5}, {y, 
.00000001, 
        1}, {z, 0, 1}, MaxRecursion -> 0, PlotPoints -> 30]\)],
"Input"],

Cell[BoxData[
    \(ContourPlot3D[
      Evaluate[\(f[x, y, z]\)[\([2]\)]], {x,  .0000001, 1.5}, {y, 
.00000001, 
        1}, {z, 0, 1}, MaxRecursion -> 0, PlotPoints -> 30]\)],
"Input"],

Cell[BoxData[
    \(ContourPlot3D[
      Evaluate[\(f[x, y, z]\)[\([3]\)]], {x,  .0000001, 1.5}, {y, 
.00000001, 
        1}, {z, 0, 1}, MaxRecursion -> 0, PlotPoints -> 30]\)],
"Input"],

Cell[BoxData[
    \(Show[%, %%, %%%]\)], "Input"],

Cell[TextData[{
  "Looks like there is no solution for ",
  Cell[BoxData[
      \(TraditionalForm\`z > 0\)]],
  ". So the above solutions should be all (real solutions)." }], "Text"]
},
FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 1024},
{0, 740}}, WindowSize->{496, 632},
WindowMargins->{{150, Automatic}, {Automatic, 9}} ]


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