Abstract

Exponential divided differences arise in numerical linear algebra, matrix-function evaluation, and quantum Monte Carlo simulations, where they serve as kernel weights for time evolution and observable estimation. Efficient and numerically stable evaluation of high-order exponential divided differences for dynamically evolving node sets remains a significant computational challenge. We present a Chebyshev-polynomial-based algorithm that addresses this problem by combining the Chebyshev-Bessel expansion of the exponential function with a direct recurrence for Chebyshev divided differences. The method achieves a computational cost of , where is the divided-difference order and is the Chebyshev truncation length. We show that scales linearly with the spectral width through the decay of modified Bessel coefficients, while the dependence on enters only through structural polynomial constraints. We further develop an incremental update scheme for dynamic node sets that enables the insertion or removal of a single node in time when the affine mapping interval is held fixed. A full \texttt{C++} reference implementation of the algorithms described in this work is publicly available.

Date

December 28, 2025

Authors

Itay Hen

Journal

arXiv preprint arXiv:2512.23061