Abstract We have developed a symbolic algebra approach to automatically produce, verify, and optimize computer code for the Fast Multipole Method (FMM) operators. This approach allows for flexibility in choosing a basis set and kernel, and can generate computer code for any expansion order in multiple languages. The procedure is implemented in the publicly available Python program Mosaic . Optimizations performed at the symbolic level through algebraic manipulations significantly reduce the number of mathematical operations compared with a straightforward implementation of the equations. We find that the optimizer is able to eliminate 20%–80% of the floating-point operations and for the expansion orders p ≤ 10 it changes the observed scaling properties. We present our approach using three variants of the operators with the Cartesian basis set for the harmonic potential kernel 1 ∕ r , including the use of totally symmetric and traceless multipole tensors.

中文翻译：

---

一种生成快速多极法算子的优化符号代数方法

摘要 我们开发了一种符号代数方法来自动生成、验证和优化快速多极法 (FMM) 算子的计算机代码。这种方法允许灵活地选择基组和内核，并且可以为多种语言的任何扩展顺序生成计算机代码。该过程在公开可用的 Python 程序 Mosaic 中实现。与方程的直接实现相比，通过代数操作在符号级别执行的优化显着减少了数学运算的数量。我们发现优化器能够消除 20%–80% 的浮点运算，并且对于扩展阶 p ≤ 10，它会改变观察到的缩放属性。