High-order derivatives of Green's functions are a key ingredient in Taylor-based fast multipole methods, Barnes-Hut $n$-body algorithms, and quadrature by expansion (QBX). In these settings, derivatives underpin either the formation, evaluation, and/or translation of Taylor expansions. In this article, we provide hybrid symbolic-numerical procedures that generate recurrences to attain an $O(n)$ cost for the the computation of $n$ derivatives (i.e. $O(1)$ per derivative) for arbitrary radially symmetric Green's functions. These procedures are general--only requiring knowledge of the PDE that the Green's function solves. We show that the algorithm has controlled, theoretically-understood error. We apply these methods to the method of quadrature by expansion, a method for the evaluation of singular layer potentials, which requires higher-order derivatives of Green's functions. In doing so, we contribute a new rotation-based method for target-specific QBX evaluation in the Cartesian setting that attains dramatically lower cost than existing symbolic approaches. Numerical experiments support our claims of accuracy and cost.

翻译：格林函数的高阶导数是基于泰勒展开的快速多极方法、Barnes-Hut $n$ 体算法以及扩展求积法（QBX）中的关键组成部分。在这些设定下，导数支撑着泰勒展开的形成、求值与/或平移。本文提出了一种混合符号数值程序，该程序生成递推关系，对于任意径向对称的格林函数，能以 $O(n)$ 的代价计算 $n$ 个导数（即每个导数 $O(1)$）。这些程序具有通用性——仅需知道格林函数所满足的偏微分方程。我们证明了该算法具有可控且理论上可理解的误差。我们将这些方法应用于扩展求积法（一种用于求解奇异层势的方法），该方法需要格林函数的高阶导数。在此过程中，我们贡献了一种新的基于旋转的笛卡尔坐标系下目标特定 QBX 求值方法，其代价显著低于现有符号方法。数值实验支持了我们关于精度和代价的论断。