Computing Simple Multiple Zeros of Polynomial Systems

TL;DR

This paper introduces methods to accurately identify, certify, and efficiently compute simple multiple zeros of polynomial systems, including bounds on zero proximity and convergence guarantees for modified Newton iterations.

Contribution

It provides new bounds on the distance between simple multiple zeros and other zeros, and develops modified Newton methods with proven convergence for such zeros.

Findings

Derived a lower bound on the minimal distance between zeros.

Proposed a numerical criterion to certify the number of zeros in a small neighborhood.

Proved quadratic convergence of modified Newton iterations for simple multiple zeros.

Abstract

Given a polynomial system f associated with a simple multiple zero x of multiplicity {\mu}, we give a computable lower bound on the minimal distance between the simple multiple zero x and other zeros of f. If x is only given with limited accuracy, we propose a numerical criterion that f is certified to have {\mu} zeros (counting multiplicities) in a small ball around x. Furthermore, for simple double zeros and simple triple zeros whose Jacobian is of normalized form, we define modified Newton iterations and prove the quantified quadratic convergence when the starting point is close to the exact simple multiple zero. For simple multiple zeros of arbitrary multiplicity whose Jacobian matrix may not have a normalized form, we perform unitary transformations and modified Newton iterations, and prove its non-quantified quadratic convergence and its quantified convergence for simple triple zeros.