The input is a system 
of n multivariate polynomials
in n unknowns,
, 
.
The output of our solver consists of root counts and
approximations to all the isolated roots of 
.
In Figure 3, the four stages of the solver are displayed. The aim of the pre-processing stage is to bring the system in a form more suitable to homotopy continuation. In the second stage, either a product homotopy (based on Bézout's theorem), or a coefficient homotopy (based on Bernshtein's theorem) can be constructed. The tuning of continuation parameters and path following by means of predictor-corrector methods is performed in the third stage. The post-processing stage consists in the validation of the computed results. Basic validation includes for instance the computation of local condition numbers whereas more elaborate validation procedures eventually require continuation themselves.
Figure 3: The four stages in the solver.
Because of the root counting stage, the user immediately has an idea of the amount of computational work that is required to solve the problem. It suffices to multiply the root count with the estimated time needed to follow one solution branch.
An important difference in Figure 3 compared with the CONSOL-logic (see [15, Chapter 4,]) is the second stage, which is not only not well elaborated in CONSOL, but is also too closely related to the continuation stage. The design of HOMPACK has the same drawback, the package gives the user no support when another start system, other than the one based on the total degree, needs to be used.
For small polynomial systems, the four stages should be very well integrated and customized. For large applications, each stage has to be clearly separated in order to have several stepping stones. Therefore, we have added options to the program, so that it can be invoked via tools which address only one particular stage of the solver.