Adaptative Step Size Selection for Homotopy Methods to Solve Polynomial Equations

Given a C1 path of systems of homogeneous polynomial equations ft, t ∈ [a, b] and an approximation xa to a zero ζa of the initial system fa, we show how to adaptively choose the step size for a Newton based homotopy method so that we approximate the lifted path (ft, ζt) in the space of (problems, solutions) pairs. The total number of Newton iterations is bounded in terms of the length of the lifted path in the condition metric.