论文标题
奇异的Euler-Maclaurin扩展
Singular Euler-Maclaurin expansion
论文作者
论文摘要
我们介绍了奇异的Euler--Maclaurin膨胀，这是一种有效计算大量奇异总和的新方法，该方法出现在浓缩物质和量子物理学中的远程相互作用系统中。与传统的Euler--Maclaurin求和公式相反，新方法也适用于可区分函数和奇异性的乘积。对于合适的非单个函数，我们表明近似误差在扩展顺序和多个多项式上以非单词函数的特征长度尺度呈指数衰减，并在提供精确的误差估计中。总和仅由积分和差分运算符近似，该操作员仅作用于非单明功能因子。此外，奇异性还包括在形成差异操作员系数的伯努利多项式的概括中。我们通过将其应用于宏观的一维晶体内部（$ 2 \ times 10^{10} $颗粒的宏观的非线性长距离力）来证明奇异欧拉蛋白扩展的数值性能。在线提供了Mathematica中的参考实现。
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.
