A new deflation method for verifying the isolated singular zeros of polynomial systems

Abstract

In this paper, we develop a new deflation technique for refining or verifying the isolated singular zeros of polynomial systems. Starting from a polynomial system with an isolated singular zero, by computing the derivatives of the input polynomials directly or the linear combinations of the related polynomials, we construct a new system, which can be used to refine or verify the isolated singular zero of the input system. In order to preserve the accuracy in numerical computation as much as possible, new variables are introduced to represent the coefficients of the linear combinations of the related polynomials. To our knowledge, it is the first time that considering the deflation problem of polynomial systems from the perspective of the linear combination. Some acceleration strategies are proposed to reduce the scale of the final system. We also give some further analysis of the tolerances we use, which can help us have a better understanding of our method. The experiments show that our method is effective and efficient. Especially, it works well for zeros with high multiplicities of large systems. It also works for isolated singular zeros of non-polynomial systems.

A Heuristic Method for Certifying Isolated Zeros of Polynomial Systems

“…Second, based on our work [35] and the work of Mourrain et al [21,36], as an application of our method, we give a heuristic method for certifying not only the isolated singular zeros of polynomial systems, but also the multiplicity structures of the isolated singular zeros of polynomial systems.…”

“…Recently, Cheng et al [35] proposed a new deflation method to reduce the multiplicity of an isolated singular zero of a polynomial system to get a final system, which owns the isolated singular zero of the input system as a simple one. Different from the previous deflation methods, they considered the deflation of isolated singular zeros of polynomial systems from the perspective of linear combination.…”

Certified numerical real root isolation for bivariate nonlinear systems

“…If we choose properly some point(s) in each suspect box in the cluster of boxes as start point(s) for Newton's method for the system, we may get the root(s) inside the cluster of boxes and remove the redundant boxes. For the derived box(es) after computing with Newton's method, we can do only a heuristic verification of a suspect box by deflation methods (see [11,21,35] and the methods mentioned therein). Notice that we may miss some root(s) or get more roots with this operation.…”

Symbolic-numeric methods oncomputing isolated singular solutions of algebraic systems

“…此外, Dian 和 Kearfott [62] 、Kearfott 和 Dian [63] 和 Kearfott 等 [64] 提出了一种基于验证非零拓扑 次数来验证非线性系统一簇解的计算方法. Cheng 等 [65] 提出了新的收缩方法来验证多项式系统奇异 解的存在性.…”