Implications of Stahl's Theorems to Holomorphic Embedding Pt. 2: Numerical Convergence

TL;DR

This paper investigates the numerical convergence of the Holomorphic Embedding Method (HEM) in power-flow analysis, highlighting the impact of Padé approximant properties and embedding choices on convergence guarantees.

Contribution

It demonstrates that finite-precision computations may not always converge and links convergence behavior to the function's branch-point topology and embedding strategies.

Findings

Finite-precision Padé approximants may fail to converge.

Convergence depends on the location of branch-points and branch-cut topology.

Poorly chosen embeddings can hinder 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.