2011/02/03 by Barry H. Dayton, Tien-Yien Li, Zhonggang Zeng · 77 citations
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic number #Algorithm #Applied mathematics #Commutative Algebra and Its Applications #Computation #Mathematical analysis #Mathematics #Multiplicity (mathematics) #Nonlinear system #Polynomial and algebraic computation #cs.NA #math.NA #msc:65H10 #msc:68W30
paper · pdf · doi:10.1090/s0025-5718-2011-02462-2
published in Mathematics of Computation 80(276), 2143-2168 (American Mathematical Society)
openalex publication_date 2011/02/03 · arxiv created 2021/03/09 · arxiv updated 2021/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.