A certified iterative method for isolated singular roots
A certified iterative method for isolated singular roots
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DOI：
10.1016/j.jsc.2022.08.006
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发表时间：
2022-08
期刊：
影响因子：
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通讯作者：
Angelos Mantzaflaris;B. Mourrain;Á. Szántó
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文献类型：
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作者：
Angelos Mantzaflaris;B. Mourrain;Á. Szántó
本文给出了一种新的证明邻域多项式系统有奇异孤立根的方法，并计算了它的重数结构。更准确地说，给定一个多项式系统f=(f 1，…，f N)∈C[x 1，…，xn]N上的牛顿迭代，在正则性条件下局部收敛到f的一个小变形，使得这个变形系统有一个精确的奇异根。迭代同时收敛到奇异根的坐标和描述根处多重数结构的所谓逆系统的系数。我们使用α理论检验来证明二次收敛，并给出了变形的大小和逼近误差的界。该方法依赖于对准时希尔伯特方案的分析，我们为其提供了新的描述。特别地，我们证明了它的一些层可以被合理地参数化，并在认证中利用这些参数化。我们在数值实验中展示了如何计算近似逆系统作为牛顿迭代的起点，以及如何快速地数值收敛到具有多重结构的奇异根，这是由我们的准则所证明的。
In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root and we compute its multiplicity structure. More precisely, given a polynomial system f=(f 1,…, f N)∈ C [x 1,…, x n] N, we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of f such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so-called inverse system that describes the multiplicity structure at the root. We use α-theory test to certify the quadratic convergence, and to give bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria.