Implications of Stahl's theorems to holomorphic embedding part II: Numerical convergence

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 the HEM and the numerical properties of a Pade approximant algorithm. We show that even if the point of interest is within the convergence domain, numerical convergence of the sequence of Pade approximants computed with finite precision is not guaranteed. We propose a convergence factor equation that can be used to both estimate the convergence rate and the capacity of the branch cut. We also show that the study of convergence properties of the Pade approximant is the study of the location of the branch-points of the function, which in turn dictate branch-cut topology and capacity and, therefore, convergence rate.