Multiple zeros of nonlinear systems

TL;DR

This paper develops a new algebraic framework and algorithms for accurately computing the multiplicity of isolated zeros in nonlinear systems, bridging numerical analysis and algebraic geometry.

Contribution

It introduces a novel multiplicity definition for nonlinear systems, along with algorithms and a depth-deflation method for precise computation of multiple zeros.

Findings

Algorithms accurately compute multiplicity and zeros with floating point arithmetic.

The multiplicity concept aligns with algebraic geometry for polynomial systems.

Fundamental properties of multiplicity are established and proved.

Abstract

As an attempt to bridge between numerical analysis and algebraic geometry, this paper formulates the multiplicity for the general nonlinear system at an isolated zero, presents an algorithm for computing the multiplicity structure, proposes a depth-deflation method for accurate computation of multiple zeros, and introduces the basic algebraic theory of the multiplicity. Furthermore, this paper elaborates and proves some fundamental properties of the multiplicity, including local finiteness, consistency, perturbation invariance, and depth-deflatability. As a justification of this formulation, the multiplicity is proved to be consistent with the multiplicity defined in algebraic geometry for the special case of polynomial systems. The proposed algorithms can accurately compute the multiplicity and the multiple zeros using floating point arithmetic even if the nonlinear system is perturbed.