2020/03/16 by Abhinav Dronamraju, Dronamraju, Abhinav, Songyan Li +11
Computer Science · Engineering · Mathematics · #Electromagnetic Simulation and Numerical Methods #FOS: Electrical engineering #Fractional Differential Equations Solutions #Matrix Theory and Algorithms #Systems and Control (eess.SY) #cs.SY #eess.SY #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.2003.07457
openalex publication_date 2020/03/16 · arxiv created 2020/03/18 · arxiv updated 2020/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.