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.

翻译：本文试图将数字分析与代数几何联系起来,将一般非线性系统的多重性表述为孤立的零,提出计算多重结构的算法,提出精确计算多重零的深度反通货膨胀方法,并介绍多重性的基本代数理论。此外,本文阐述并证明多重性的一些基本特性,包括局部的有限性、一致性、易变性和深度减缩性。作为这一表述的理由,多元系统的特殊情况,其多重性被证明符合代数性几何性所定义的多重性。即使非线性系统处于交错状态,拟议的算法也可以精确地用浮点算法计算多重性和多零性。