Stahl定理对全纯嵌入的意义。2：数值收敛
Implications of Stahl's Theorems to Holomorphic Embedding Pt. 2: Numerical Convergence
摘要：
在电力工程界被称为Stahl定理的东西已经被用来证明全纯嵌入法（HEM）的收敛保证，因为它适用于电力流问题。在这篇论文的第二部分中，我们研究了HEM的数值收敛性和Padé近似算法的数值特性。我们表明，即使收敛域是相同的功能的域，数值收敛的序列的Padé逼近计算有限精度是不能保证的。我们还表明，Padé逼近的收敛性的研究是研究的分支点的功能，这决定了分支切割拓扑结构和能力，因此，收敛速度的位置。我们展示了如何选择不好的嵌入可以防止数值收敛。
英文摘要：
What has become known as Stahl's Theorem in power-engineering circles has been used to justify a convergence guarantee of the Holomorphic Embedding Method (HEM) as it applies to the power-flow problem. In this, the second part of a two-part paper, we examine implications to numerical convergence of HEM and the numerical properties of a Padé approximant algorithm. We show that even if the convergence domain is identical to the function's domain, numerical convergence of the sequence of Padé approximants computed with finite precision is not guaranteed. We also show that the study of convergence properties of the Padé approximant is the study of the location of branch-points of the function, which dictate branch-cut topology and capacity and, therefore, convergence rate. We show how poorly chosen embeddings can prevent numerical convergence.
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