How to solve a non linear ovedetermined system of equations??

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Alphonce Owayo on 28 Feb 2021
Commented: Alphonce Owayo on 4 Mar 2021
eq(k)=(sqrt(x1-x0)^2+(y1-y0)^2+(z1-z0)^2)==c*(t1-t0); Unknowns are x0 y0 z0 and t0, eqn(k) are 8. I am not exactly sure what tricks to unleash with the optimization algorithms in matlab. I have even tried to simplify the equation further to sqrt (a1^2+a2^2+a3^2+a4^2==c*(ti-t0)); hoping to simplify it and solve for a1 a2 a3 and a4, find the sqrt and hopefully give a good estimate for the unknowns at this point I am frustrated its not working. I have watched all the optimization videos,they make really look easy but it isnt, someone help please before I loose my mind, thanks.
Alphonce Owayo on 28 Feb 2021
The equation is just of that one form 8 times ofcourse with varying values of X1 Y1 Z1 and ti. I hope it's clearer.

Bjorn Gustavsson on 28 Feb 2021
You can always try with regular minimization. Perhaps something like this:
function err = err_1(pars,x,y,z,y)
c = 3; % guess, perhaps close to some factor of 10 off?
x0 = pars(1);
y0 = pars(2);
z0 = pars(3);
t0 = pars(4);
err = sum(((sqrt(x-x0).^2 + (y-y0).^2 + (z-z0).^2) - c*(t-t0)).^2);
end
Then you'll have to make an initial guess - that should preferably be close to the solution, this might be more or less important depending on the function (I haven't thought about what would be the case for your...);
x0y0z0t0_0 = [1,2,3,4]; % you hopefully have a hunch..
x0y0z0t0 = fminsearch(@(pars) err_1(pars,x,y,z,y),x0y0z0t0_0)
You can also use lsqnonlin if fminsearch is too slow. But then you'll have to modify err_1 to return reiduals instead of the sum of the residuals squared.
HTH
Alphonce Owayo on 4 Mar 2021
Sorry I misspelled ' lsqnonlin'Yeah you are right, and with lsqnonlin you can restrict the search so you pretty reasonable values.

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