Note on the end game in homotopy zero curve tracking

Abstract

Homotopy algorithms to solve a nonlinear system of equations f(x) = 0 involve tracking the zero curve of a homotopy map p(a, λ, x) from λ = 0 until λ = 1. When the algorithm nears or crosses the hyperplane λ = 1, an “end game” phase is begun to compute the solution x¯ satisfying p(a, λ, x¯) = f(x¯) = 0. This note compares several end game strategies, including the one implemented in the normal flow code FIXPNF in the homotopy software package HOMPACK.