PreprintCC BY-NC-ND 4.0HAL
ROOT ISOLATION FOR ANALYTIC FUNCTIONS USING CUBIC HERMITE INTERPOLATION.
Abstract
We present a root isolation algorithm for an analytic function. The method combines interval subdivision with a heuristic test based on local cubic Hermite interpolation. Whenever the function is sufficiently close to its cubic Hermite interpolant at a few sample points, we assume that the critical points of the cubic interpolant give a valid root isolation for the original function. Such an algorithm cannot, in general, provide a rigorous certification of the computed intervals. Nevertheless, extensive numerical experiments show that it is highly efficient and almost always produces the correct results. We hope that a probabilistic or exact algorithm can be derived from this work.
Related
- Exact barycentric rational interpolation of analytic functions with an application to the Bessel functions
- A Direct Approach to Hermite Interpolation
- Conjectures of Bernstein and Erdős for weighted Lagrange interpolation with Hermite weight
- Arnoldi-Enhanced Multivariate Hermite Interpolation of Manifold-Valued Data
- A counterexample to vertex interpolation by planar C1 cubic splines
BibTeX
@misc{raffalli2026root,
title = {{ROOT ISOLATION FOR ANALYTIC FUNCTIONS USING CUBIC HERMITE INTERPOLATION.}},
author = {Christophe Raffalli},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2610.02934},
eprint = {2610.02934},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05765782v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - ROOT ISOLATION FOR ANALYTIC FUNCTIONS USING CUBIC HERMITE INTERPOLATION.
AU - Christophe Raffalli
PY - 2026
DA - 2026/09/26
PB - HAL
DO - 10.48550/arxiv.2610.02934
UR - https://hal.science/hal-05765782v1
AB - We present a root isolation algorithm for an analytic function. The method combines interval subdivision with a heuristic test based on local cubic Hermite interpolation. Whenever the function is sufficiently close to its cubic Hermite interpolant at a few sample points, we assume that the critical points of the cubic interpolant give a valid root isolation for the original function. Such an algorithm cannot, in general, provide a rigorous certification of the computed intervals. Nevertheless, extensive numerical experiments show that it is highly efficient and almost always produces the correct results. We hope that a probabilistic or exact algorithm can be derived from this work.
ER -