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The Four Stages of the Solver

The root count provides important information about 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 path.

In Figure 4, the four stages of the solver are displayed.

Figure 4: The four stages in the solver.


The aim of preconditioning is to bring the system in a form more suitable to homotopy continuation. In the second stage, a root-counting method is applied to construct a start system. 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.

Jan Verschelde