How can I choose the right numerical solution from NEQNF?

Question

I'm using a function (NEQNF manual page here) which I call using

call neqnf(SYSTEM_OF_EQUATIONS, x, xguess=x_GUESS, itmax = 10000)

where SYSTEM_OF_EQUATIONS is the subroutine that contains equations

f(1)=...x(2)...x(1)...
f(2)=...x(1)...x(4)...
f(3)=...x(3)...x(4)...
f(4)=...x(1)...x(5)...
f(5)=...x(1)...x(5)...

from IMSL libraries on Fortran that lets me to solve a non-linear system with five unknowns in five equations. Because there exists more than one solution (couple of five numbers, real or complex, that solve my system), how can I choose which couple to "use" as solution?

I link an online solver with already entered a piece of my system (only two unknowns in two equations, other variables are constant in this example) which easily show you that there exists more than one solution.

example

To conclude my issue I can say that I have to choose the couple of variables which let other variables to be positive, so an easy check is the way to choose the couple.

Answer 1

I don't think the question has anything to do with programming, but I will show how I understand the problem.

You supply an initial guess. Then the method just converges to some solution by a modification of a Newton method.

You can choose the root by the placement of the initial guess. However, the convergence pattern can be very unpredictable (even fractal - https://en.wikipedia.org/wiki/Newton_fractal ) and it may be very difficult to choose the particular root using the initial guess.