An a posteriori certification algorithm for Newton homotopies

Abstract

A Newton homotopy is a homotopy that involves changing only the constant terms. They arise naturally, for example, when performing monodromy loops, moving end effectors of robots, and simply when trying to compute a solution to a square system of equations. Previous certified path tracking techniques have focused on using an a priori certified tracking scheme which means that the stepsize is constructed so that the result automatically satisfies some conditions. These schemes use pessimistic stepsizes that can be much smaller than those used by heuristic tracking methods. This article designs an a posteriori certification scheme that uses the result of a heuristic tracking scheme as input to produce a certificate that the path was indeed tracked correctly, e.g., no path jumpings occurred. By using an a posteriori approach, each step can be certified independently and thus certification of the path can be performed in parallel. Examples are presented demonstrating the efficiency of this a posteriori certification approach.