Calculating the divided differences of the exponential function by addition and removal of inputs

TL;DR

This paper presents a novel efficient method for updating divided differences of the exponential function using input addition and removal, enabling handling of much longer input lists than existing methods, with applications in quantum Monte Carlo algorithms.

Contribution

The authors introduce a new identity-based algorithm for updating exponential divided differences efficiently upon input modifications.

Findings

Re-evaluation of divided differences requires only O(s n) operations and memory.

The method handles input lists much longer than current state-of-the-art.

Application demonstrated in quantum Monte Carlo weight calculations.

Abstract

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.