The way the nonlinear system of equations is generally solved is by a method based on Newton Methodology (Newton-Raphson, Newton relaxation, gradient procedures for optimization, predictor-corrector, etc). One of the reasons for this widespread use of Newton-like methods is that generally a good starting point can be given by experimental observations.
It is important to quote the CCETT case where the design of a new filter bank was accomplished by using Gb and RealSolving, two packages developed by members (J.-C. Faugere and F. Rouillier) of the FRISCO consortium and which will be integrated in the FRISCO framework.
There are two possibilities for the time to be spent on the resolution of the nonlinear system under consideration: the solution must be available in real time (in the inverse kinematics problem for a manipulator) or more time is allowed (when fitting the parameters of the theoretical model according to the experimental data available).
The solutions to be computed are always real (with the exception of the EdF example) and the best accuracy of the answers is around 10-7. This arises from the design of a variational CAD/CAE system at LABEIN where the coefficients were exact rational numbers. It is a surprising fact that, in at least two cases (tolerancy analysis at LABEIN and simulation at TECNATOM), the real-world problem implied that the particular nonlinear system of equations had only one real solution.