A superlinear arclength endgame for corank-one singular polynomial systems

Abstract

In Numerical Algebraic Geometry (NAG) isolated solutions of polynomial systems are usually computed by tracking a solution curve defined by a homotopy equation. The tracking problem becomes especially challenging close to a singular root (the ``endgame'' regime). Existing approaches include power series endgames, Cauchy endgames, and various methods that regularize the system via dual-space-based {\em deflation}. We introduce a new ``Arclength Endgame'' for corank-1 systems which combines the idea of the classical {\em pseudo-arclength continuation method} with the estimation of the Puiseux series of the curve. We formally prove that it has an R-superlinear rate of convergence in some neighborhood of the root. The method uses only evaluations of the system and its Jacobian. In contrast, previous homotopy techniques with proven superlinear convergence (such as deflation) require computing additional derivatives of the system, or need additional nondegeneracy assumptions involving higher-order derivatives of the system at the singular root. A key step in our approach (as well as in the standard power series endgame) is estimating the ratio $k_1/c$ of the Puiseux series of the curve. We present a new method for that, and discuss its advantages over previous methods.