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

TL;DR

The paper addresses locating singularities at the ends of solution paths in polynomial homotopy continuation. It couples numerical analytic continuation with extrapolation methods such as Padé, Aitken, and the rho algorithm, integrated via PHCpack and the Python interface phcpy to enable step-by-step path tracking with a priori step-size control. Key contributions include demonstrations on test homotopies with near and far poles (e.g., singularities at $t=1$ and $t=1/2 + i$, and a triple root at $(x,y)=(1,2)$), provision of iterated Aitken extrapolation that yields substantial accuracy gains, and practical open-source tooling with installation workflows. The work significantly improves robustness and accessibility for locating singular endpoints in polynomial systems, enhancing numerical homotopy continuation workflows.

Abstract

PHCpack is a software package for polynomial homotopy continuation, which provides a robust path tracker [Telen, Van Barel, Verschelde, SISC 2020]. This tracker computes the radius of convergence of Newton's method, estimates the distance to the nearest path, and then applies Padé approximants to predict the next point on the path. A priori step size control is less sensitive to finely tuned tolerances than a posteriori step size control, and is therefore robust. The Python interface phcpy is extended with a new step-by-step tracker and is applied to experiment with extrapolation methods to accurately locate the singular points at the end of solution paths.

Figures (5)

• Figure 1: A phase portrait of $f(z) = \sqrt{4/5(1/2-I)(1-z)(1/2+I-z)}$, depicting at the left the color-coded phase $f/|f|$ on the domain of $f$. At the right is a polar chessboard.
• Figure 2: At the left, the pole $P = 1/2 + I$ is relatively close to 1, while at the right, the pole $P = -4 + 16 I$ is much farther from 1.
• Figure 3: Points on one solution path starting at $(1, 1)$ and converging to $(1, 2)$.
• Figure 4: A phase portrait of $f(z)$, defined in (\ref{['eqojika1poles']}) by the poles of one path converging to a singular solution.
• Figure 5: At each point on the path, a circle is drawn, with center at the point and the radius is the distance to the predicted solution.