Exact root isolation without missed roots

Establish whether, for every analytic function with no multiple roots, there exist parameters $\varepsilon\in(0,1)$ and $n\in\mathbb{N}^*$ such that the cubic-Hermite-interpolation root isolation algorithm never misses any roots, and determine whether an exact version of the algorithm can be obtained.

Background

The paper proves termination and guarantees that every returned interval contains at least one root, but it does not prove that all roots are found. The stated conjecture asks whether suitable values of the quality threshold ε\varepsilon and the number of regularly spaced test points nn prevent missed roots for every analytic function with no multiple roots. It also explicitly identifies the existence of an exact algorithm as unknown, while noting that experiments suggest relatively small parameter values work for many polynomial families.

References

For any analytic function with no multiple roots, we can find $\varepsilon \in ]0,1[$ and $n \in \mathbb{N}*$ such that the above algorithm misses no roots (never or with very low probability, we do not know if we can get an exact algorithm).

— Root isolation for analytic functions using cubic hermite interpolation  (2610.02934 - Raffalli, 2 Oct 2026) in Conjecture following the termination proposition, Section "Description of the Algorithm"