Implications of Stahl's Theorems to Holomorphic Embedding Pt. 2: Numerical Convergence
Abstract
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 Pad\'e approximant algorithm. We show that even if the convergence domain is identical to the function's domain, numerical convergence of the sequence of Pad\'e approximants computed with finite precision is not guaranteed. We also show that the study of convergence properties of the Pad\'e 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.
Keywords
Cite
@article{arxiv.2003.07457,
title = {Implications of Stahl's Theorems to Holomorphic Embedding Pt. 2: Numerical Convergence},
author = {Abhinav Dronamraju and Songyan Li and Qirui Li and Yuting Li and Daniel Tylavsky and Di Shi and Zhiwei Wang},
journal= {arXiv preprint arXiv:2003.07457},
year = {2020}
}