Padé逼近的根分布及其对全纯嵌入法收敛性的影响
Root Distribution in Padé Approximants and its Effect on Holomorphic Embedding Method Convergence
摘要：
在电力电力系统仿真领域中，普遍存在求解非线性代数方程组的需求。虽然基于牛顿的方法已被用于优势，但它们有时不会收敛，让用户怀疑是否存在解决方案。除了提高鲁棒性之外，全纯嵌入方法（HEM）的一个优点是，即使它们不收敛，逆$α$平面中函数的Padé逼近（PA）的根图也可以用于确定解是否存在。应用于围绕原点展开的函数的近对角PA的收敛因子（CF）由相关联的分支切割（BC）的对数容量和评估点距原点的距离确定。然而，控制这一比率的基本机制一直不清楚。我们证明了在复平面上BC上的PA根的最终分布弱收敛于在物理设置中具有相同拓扑结构的2-D导体系统上的静电荷的平衡分布。这一点，随着马克劳林级数的性质可以用来解释CF方程的结构，我们证明了理论的收敛性行为，与数值实验。
英文摘要：
The requirement for solving nonlinear algebraic equations is ubiquitous in the field of electric power system simulations. While Newton-based methods have been used to advantage, they sometimes do not converge, leaving the user wondering whether a solution exists. In addition to improved robustness, one advantage of holomorphic embedding methods (HEM) is that, even when they do not converge, roots plots of the Padé approximants (PAs) to the functions in the inverse-$α$ plane can be used to determine whether a solution exist. The convergence factor (CF) of the near-diagonal PAs applied to functions expanded about the origin is determined by the logarithmic capacity of the associated branch cut (BC) and the distance of the evaluation point from the origin. However the underlying mechanism governing this rate has been obscure. We prove that the ultimate distribution of the PA roots on the BC in the complex plane converges weakly to the equilibrium distribution of electrostatic charges on a 2-D conductor system with the same topology in a physical setting. This, along with properties of the Maclaurin series can be used to explain the structure of the CF equation We demonstrate the theoretical convergence behavior, with numerical experiments.
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