Vali: Validation of the computations

We distinguish between three different types of validation of the results. They are listed according to increasing order of complexity:

1. Newton's method for refining. A path leads to an approximate solution if it can be refined up to machine precision, within quadratic convergence of Newton's method. Estimates for the condition number of the Jacobian matrix are computed as well. Additionally, Vali checks whether all solutions are different from each other and counts the real roots.

2. Path directions to face normals. The computed path directions are verified against the faces of the polynomial system. Path failures correspond to solutions at infinity if the directions of the path lead to an outer normal of a face, for which every equation has at least two monomials.

3. Winding numbers. Computing winding numbers is an example of an exhaustive end game strategy, proposed in [16]. By continuation, solution paths are winded around the solution, which can be approximated more accurately by averaging. Our implementation provides a primitive facility for computing winding numbers.

Homotopy continuation methods are very good in delivering paths to solutions. The lack of an adequate end game procedure might spoil all the advantages of the solver. However, according to standard numerical practise, the user should participate actively when faced to ill-conditioned problems. The program detects singularities and solutions at infinity, whereafter the user has to decide whether special techniques must be applied, or whether these ill-conditioned solutions are spurious to the application and merit no further attention.