We present the singular Euler--Maclaurin expansion, a new method for the efficient computation of large singular sums that appear in long-range interacting systems in condensed matter and quantum physics. In contrast to the traditional Euler--Maclaurin summation formula, the new method is applicable also to the product of a differentiable function and a singularity. For suitable non-singular functions, we show that the approximation error decays exponentially in the expansion order and polynomially in the characteristic length scale of the non-singular function, where precise error estimates are provided. The sum is approximated by an integral plus a differential operator acting on the non-singular function factor only. The singularity furthermore is included in a generalisation of the Bernoulli polynomials that form the coefficients of the differential operator. We demonstrate the numerical performance of the singular Euler--Maclaurin expansion by applying it to the computation of the full non-linear long-range forces inside a macroscopic one-dimensional crystal with $2\times 10^{10}$ particles. A reference implementation in Mathematica is provided online.

翻译：我们展示了单一的 Euler-Maclaurin 扩展, 这是在浓缩物质和量子物理学的长距离互动系统中有效计算大单价的新方法。 与传统的 Euler- Maclaurin 校验公式相反, 新方法也适用于不同函数和单数的产物。 对于合适的非星系函数, 我们显示近似错误在扩展顺序中以指数方式递减, 以非星系函数的特性长度尺度( 提供精确的误差估计) 中以多元方式递增, 以一个整体加一个差运算者仅对非星系函数进行计算, 与传统的 Euler- Maclaurin 相较。 我们通过将它应用于计算一个宏观的单层单层单层晶体中的全非线长强度( 2\ 10 美元 ) 10 的计算方法, 来显示单一的 Euler- Maclaurin 扩展的数值性能。 在 Mathematica 中提供在线执行的参考数据 。