Before revision:  The algorithm removes standard restrictions for path following FoV methods that generally cannot deal with decomposing matrices A due to possible eigencurve crossings for such A . Tests and numerical comparisons are included   .
After revision:  The algorithm removes standard restrictions for path following FoV methods that generally cannot deal with decomposing matrices A due to possible eigencurve crossings for such A . Tests and numerical comparisons are included . Our ZNN based method is coded for sequential and parallel computations and both versions run very accurately and very fast when compared with Johnson's global QR eigenvalue and Bedixon rectangle based method that computes complete eigenanalyses of H\!K_A(t_k) for every chosen t_k \in .