Exponential divided differences via Chebyshev polynomials

TL;DR

This paper introduces a Chebyshev-polynomial-based algorithm for efficiently and stably computing high-order exponential divided differences, with applications in numerical linear algebra and quantum simulations.

Contribution

It presents a novel Chebyshev-based method combining Bessel expansion and recurrence relations, improving computational efficiency and stability for dynamic node sets.

Findings

Achieves ${ m O}(qN)$ computational cost.

Linear scaling of Chebyshev truncation with spectral width.

Efficient incremental update scheme for dynamic nodes.

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 ${\cal O}(qN)$, where $q$ is the divided-difference order and $N$ is the Chebyshev truncation length. We show that $N$ scales linearly with the spectral width through the decay of modified Bessel coefficients, while the dependence on $q$ 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 ${\cal O}(N)$ 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.