Computing singular solutions of polynomial systems: towards superlinear convergence without deflation

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 make the following contributions. (1) For corank-1 systems we introduce a new ``Arclength Endgame'' 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 a superlinear rate of convergence in some neighborhood of the root. The method uses only evaluations of the system and its Jacobian, whereas previous techniques with proven superlinear convergence (such as deflation) require computing additional derivatives of the system. (2) For systems with a larger corank we propose a heuristic ``Lifted Arclength Endgame'', which shows promising experimental results. (3) A key step in our approach (as well as in the standard power series endgame) is estimating the Puiseux series of the curve, which is characterized by fractional exponents $k_i/c$ for $i\ge 1$ together with associated coefficients. Previous work addressed only estimating the ratio $k_1/c$. We present a new method for that which empirically appears to be more stable than previous methods, and also show how to estimate $k_i/c$ for $i\ge 2$.