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.