We introduce a method for calculating the divided differences of the exponential function by means of addition and removal of items from the input list to the function. Our technique exploits a new identity related to divided differences recently derived by F. Zivcovich [Dolomites Research Notes on Approximation 12, 28-42 (2019)]. We show that upon adding an item to or removing an item from the input list of an already evaluated exponential, the re-evaluation of the divided differences can be done with only $O(s n)$ floating point operations and $O(s n)$ bytes of memory, where $[z_0,\dots,z_n]$ are the inputs and $s \propto \max_{i,j} |z_i - z_j|$. We demonstrate our algorithm's ability to deal with input lists that are orders-of-magnitude longer than the maximal capacities of the current state-of-the-art. We discuss in detail one practical application of our method: the efficient calculation of weights in the off-diagonal series expansion quantum Monte Carlo algorithm.

中文翻译：

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通过添加和删除输入来计算指数函数的除差

我们介绍了一种通过将输入列表中的项添加到函数中来计算指数函数的除差的方法。我们的技术利用了最近由 F. Zivcovich [Dolomites Research Notes on Approximation 12, 28-42 (2019)] 得出的与划分差异相关的新身份。我们表明，在已评估指数的输入列表中添加或删除项目后，可以仅使用 $O(sn)$ 浮点运算和 $O(sn) 来重新评估划分的差异$ 字节的内存，其中 $[z_0,\dots,z_n]$ 是输入，$s \propto \max_{i,j} |z_i - z_j|$。我们证明了我们的算法能够处理比当前最先进技术的最大容量长几个数量级的输入列表。