Numerical Solution to System of Non-linear Equations

  1. 1. The problem statement, all variables and given/known data

    I am having problems solving this system of non-linear equations in Matlab and Mathematica. a,b,c,d,e,f are all independent variables.

    In Matlab I use fsolve() and get one solution, in Mathematica I use NSolve[] and get another totally different solution. Mathematica also reports this error: "NSolve::infsolns: Infinite solution set has dimension at least 1."

    What is the problem here? Is it that there is no unique solution to this set of equations?

    2. Relevant equations

    [tex]
    -\frac{21 a b c}{200000}+\frac{21 a^2 d}{400000}-\frac{21 b^2 d}{400000}+\frac{e}{100}=0
    [/tex]
    [tex]
    \frac{21 a^2 c}{400000}-\frac{21 b^2 c}{400000}+\frac{21 a b d}{200000}+\frac{f}{100}=0
    [/tex]
    [tex]
    \frac{c}{100}-\frac{19 a b e}{200000}+\frac{19 a^2 f}{400000}-\frac{19 b^2 f}{400000}=0
    [/tex]
    [tex]
    \frac{d}{100}+\frac{19 a^2 e}{400000}-\frac{19 b^2 e}{400000}+\frac{19 a b f}{200000}=0
    [/tex]
    [tex]
    \frac{a}{100}+\frac{b c e}{10000}-\frac{a d e}{10000}-\frac{a c f}{10000}-\frac{b d f}{10000}-\frac{1}{5 \sqrt{2}}=0
    [/tex]
    [tex]
    \frac{b}{100}+\frac{a c e}{10000}+\frac{b d e}{10000}+\frac{b c f}{10000}-\frac{a d f}{10000}=0
    [/tex]
     
  2. jcsd
  3. Ray Vickson

    Ray Vickson 6,281
    Science Advisor
    Homework Helper

    I used a re-scaled system (which you should always do, as a matter of course): I set x = 100*X for x = a,b,c,d,e,f and X = A,B,C,D,E,F, then solved the system exactly in Maple. Here is the solution:


    > sol:=%;

    sol := {A = %1, B = 0, C = C,

    2 2 2 1/2
    D = 1/20 RootOf(_Z + 400 C + 190 %1 - 19 %1 2 ),

    2 2 2 2 1/2
    E = -21/8 %1 RootOf(_Z + 400 C + 190 %1 - 19 %1 2 ),

    2
    F = -105/2 %1 C},

    1/2
    2
    {A = ----, B = 0, C = 0, D = 0, E = 0, F = 0}, {A = %1, B = 0,
    10

    2 2 1/2
    C = 1/20 RootOf(_Z + 190 %1 - 19 %1 2 ), D = 0, E = 0,

    2 2 2 1/2
    F = -21/8 %1 RootOf(_Z + 190 %1 - 19 %1 2 )}

    4
    %1 := RootOf(9975 _Z - 4)

    It is a bit hard to read here, but there are two distinct solution types, one being
    {A = .14150988295122837392, B = 0., C = .24393682254305351184e-2*I, D = 0., E = 0., F = -.25645443231646279921e-2*I} (where I = sqrt(-1)) and the other contained in the first curly brackets. In this "other" solution, C is arbitrary and all the other variables are determined as functions of C, but of a complicated kind: C appears in the coefficients of 4th degree polynomials, and A, B, etc., involve roots of these polynomials. So, no, the solution is certainly not unique.

    RGV
     
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