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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2512.23061 |
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| _version_ | 1866911342865154048 |
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| author | Hen, Itay |
| author_facet | Hen, Itay |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23061 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exponential divided differences via Chebyshev polynomials Hen, Itay Computational Physics 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. |
| title | Exponential divided differences via Chebyshev polynomials |
| topic | Computational Physics |
| url | https://arxiv.org/abs/2512.23061 |