On isolation of simple multiple zeros and clusters of zeros of polynomial systems

Abstract

Given a polynomial system f associated with a simple multiple zero x of multiplicity µ, 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 µ 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.

Symbolic-numeric methods oncomputing isolated singular solutions of algebraic systems

“…, ρ), 且其重数为 µ = ρ + 1. 定理 4.9 [47] 矩阵 Dg(ξ, ζ, 0) 可逆. 针对单次收缩即终止的近似奇异解, 我们也可通过计算 它的局部隔离界来得到代数系统值的下界, 进而验证满足 (4.2) 的一簇解.…”

“…首先, 他们给出了 简单 2 重根的局部隔离界; 其次, 他们提出了一个基于 Rouché 定理的判定条件, 使得满足条件的近 似奇异解在某邻域内存在重数的和等于 2 的一簇解. 正是受该方法启发, 我们将代数系统一簇解的验 证方法 (4.2) 推广至 κ = 1 和单次收缩即终止的两种特例情形 (参见文献 [47,48]). 另外, Yakoubsohn [57] 将阿尔法理论推广至单变元多项式方程的重根情形, 并给出了验证其一簇解的计算方法 (参见文 献 [58]).…”

“…本文结合作者和合作者的研究成果 [13,20,21,44,47,48] , 综述了符号数值方法在计算代数方程组孤立 奇异解、特别是近似奇异解精化与验证方面的研究进展. Smale 关于解析系统非奇异解的阿尔法理论 有着非常重要的理论和应用价值, 给出了量化的 Newton 迭代二次收敛的一般准则.…”

Isolating clusters of zeros of analytic systems using arbitrary-degree inflation

“…Many numeric and symbolic algorithms struggle with computing or approximating zeros of zero-dimensional systems of polynomials that are either singular or clustered. For some algorithms, however, providing information about the clusters, such as their sizes, locations, and distances from the other zeros, can be used to restore the efficiency of these algorithms [7,10]. In addition, data about these clusters can also be used to derive more precise estimates on the algorithmic complexity of algorithms, see, for example, [14,3,4,1].…”

Locating the Closest Singularity in a Polynomial Homotopy

“…Although deflation has been addressed by many (e.g. [4], [7], [6], [12], [13], [14], [23,24], [27]), the question on when to deflate is an open problem.…”