I have more non-linear equations than unknowns. How can I solve it in matlab? One method I know is non-linear least squares, how can I implement it in Matlab or you just suggest me anything you deem more proper.

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sermet 2013-12-2

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https://ww2.mathworks.cn/matlabcentral/answers/108260-i-have-more-non-linear-equations-than-unknowns-how-can-i-solve-it-in-matlab-one-method-i-know-is-n

编辑： Matt J 2013-12-3

x1=475060;y1=1096300;z1=4670;x2=481500;y2=1094900;z2=4694;x3=482230;y3=1088430;z3=4831;x4=478050;y4=1087810;z4=4775;

% ((x-x1)^2)+((y-y1)^2)+((z-z1)^2)=5942.607^2 % ((x-x2)^2)+((y-y2)^2)+((z-z2)^2)=2426.635^2 % ((x-x3)^2)+((y-y3)^2)+((z-z3)^2)=5094.254^2 % ((x-x4)^2)+((y-y4)^2)+((z-z4)^2)=5549.874^2

%4 equations

%x y z unknowns

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Matt J 2013-12-2

Is this question still relevant? It seems to be a duplicate of your earlier question

http://www.mathworks.com/matlabcentral/answers/108000-i-have-4-equations-and-3-unknowns-for-trilateration-problem-i-wanna-solve-these-3-unknowns-from-the

and you Accepted answers given there.

sermet 2013-12-3

Fsolve cannot solve these non linear equation with least squares.

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Answer 1

Matt J 2013-12-3

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编辑：Matt J 2013-12-3

在 MATLAB Online 中打开

Aside from FSOLVE, you can try LSQNONLIN which is another least squares solver. However, I obtain nearly the same results (x,y,z) for your problem with both FSOLVE and LSQNONLIN.

    F=@(x,y,z)[((x-x1)^2)+((y-y1)^2)+((z-z1)^2)-5942.607^2 
    ((x-x2)^2)+((y-y2)^2)+((z-z2)^2)-2426.635^2; 
    ((x-x3)^2)+((y-y3)^2)+((z-z3)^2)-5094.254^2;
    ((x-x4)^2)+((y-y4)^2)+((z-z4)^2)-5549.874^2
    ]
    fun=@(u) F(u(1),u(2),u(3));
    x0=mean([x1 y1 z1;x2 y2 z2; x3 y3 z3; x4 y4 z4])
    [u1,fval1]=fsolve(fun,x0),
    [u2,~,fval2]=lsqnonlin(fun,x0)
>> norm(u1-u2)
ans =
     9.7868e-04

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sermet 2013-12-3

thanks Roger, I got it.

Matt J 2013-12-3

编辑：Matt J 2013-12-3

在 MATLAB Online 中打开

@Sermet,

FSOLVE is telling you that an exact solution could not be found. However, because you had more equations than unknowns, it did default to a least squares algorithm. So, there was no real point to using lsqnonlin, except in my example to prove to you that they produce nearly the same thing.

As the three of us have discussed before, an exact solution is generally not available. However, for the particular data you've given, the least squares solution's residuals fairly small, compared to the radii of the spheres, in all 4 equations,

>> [xyz,residuals]=fsolve(fun,x0)
xyz =
     1.0e+06 *
      0.4800    1.0930    0.0045
residuals =
      6.5293
    -10.6212
     12.1881
    -10.1086