How can I solve overdetermined systems of equations using GSL?

[csopi]

Hi,
I need to solve the following system of equations:

qa = fa (qa, qb, qc, qd)
qb = fb (qa, qb, qc, qd)
qc = fc (qa, qb, qc, qd)
qd = fd (qa, qb, qc, qd)

where all the variables and functions are complex. However, there are some additional constraints for the variables, let's call them

na (qa, qb, qc, qd) = 1
nb (qa, qb, qc, qd) = 1
nc (qa, qb, qc, qd) = 1
nd (qa, qb, qc, qd) = 1

My problem is, that if I take into account the constraints, the system will be overdetermined (4 variables, 8 equations). However, it is known, that it still has a lot of solutions.

I am quite new to GSL (GNU Scientific Library), and I do not know, how to solve overdetermined systems and I cannot find anything in the manual either.

Do you have any suggestions?

 

[Future Bruno]

One way to approach this problem is to use a numerical optimization approach. You can use an optimization technique such as the Levenberg-Marquardt algorithm to minimize the differences between the equations and the constraints. This approach will allow you to find a set of values for the variables that minimizes the difference between the equations and the constraints. This may not be the exact solution, but it will give you a good approximation. Alternatively, you could use a method such as genetic algorithms to search for a solution. This would involve setting up an algorithm which randomly generates a set of values for the variables and then evaluates how close they are to satisfying the equations and constraints. The algorithm would then modify the values until a satisfactory solution is found. Finally, you could also try using a symbolic mathematical solver, such as Mathematica or Maple, to solve the system of equations symbolically. This would involve writing a program to set up the system of equations and then calling the solver to solve it. Whichever approach you choose, it should be possible to find a solution to your system of equations.