Extrapolating Solution Paths of Polynomial Homotopies towards Singularities with PHCpack and phcpy

A robust path tracker [Telen, Van Barel, Verschelde, SISC 2020] computes the radius of convergence of Newton's method, estimates the distance to the nearest path, and then applies Pad\'e approximants to predict the next point on the path. Apriori step size control is less sensitive to finely tuned tolerances than aposteriori step size control, and is therefore robust. Extrapolation methods are effective to accurately locate the singular points at the end of solution paths, as illustrated with phcpy, the scripting interface to PHCpack.