Adaptative Step Size Selection for Homotopy Methods to Solve Polynomial Equations

TL;DR

This paper introduces an adaptive step size strategy for homotopy methods solving polynomial equations, optimizing the number of Newton iterations by considering the path's condition metric length.

Contribution

It presents a novel adaptive step size selection technique for Newton-based homotopy methods, improving efficiency in solving polynomial systems.

Findings

Bounded total Newton iterations based on path length in condition metric

Effective adaptation of step size along the homotopy path

Enhanced efficiency in polynomial equation solving

Abstract

Given a C^1 path of systems of homogeneous polynomial equations f_t, t in [a,b] and an approximation x_a to a zero zeta_a of the initial system f_a, we show how to adaptively choose the step size for a Newton based homotopy method so that we approximate the lifted path (f_t,zeta_t) 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.