Adaptive step-size selection for homotopy methods to solve polynomial equations

Abstract

Abstract. Given a C 1 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.

Fast Linear Homotopy to Find Approximate Zeros of Polynomial Systems

“…A specific description of the method has been given in [5]. Other descriptions were obtained independently and simultaneously, see [16,19]. In [6,24], a Macaulay2 implementation of the homotopy algorithm of [5] is presented.…”

“…Here we use an explicit description [5] of this method (there are other explicit descriptions available, see [16,19]). It performs a number of "homotopy steps", each of them being one application of projective Newton's method.…”

Rigid continuation paths I. Quasilinear average complexity for solving polynomial systems

“…This is called the "µ estimate". Explicit algorithms that achieve this bound have been designed by Beltrán (2011), Dedieu et al (2013), and Hauenstein and Liddell (2016). A simpler but weaker form, called the "µ 2 estimate", reads…”

An a posteriori certification algorithm for Newton homotopies

“…8] for a general overview), which provides sufficient conditions that Newton's method will quadratically converge immediately starting at a given point. The certified tracking methods of [6,7,8,10,11,23] depend on using α-theory to compute the size of the next step to prevent path jumpings whereas we will use it to certifiably determine if a path jumping occurred. The software alphaCertified [16] implements the necessary α-theoretic routines needed with [15] demonstrating its use to a posteriori prove results regarding endpoints of paths.…”