Optimality of regularly spaced interpolation test points

Determine whether choosing the additional Hermite-interpolation test points at regularly spaced locations within an interval is optimal for the root isolation algorithm.

Background

The root isolation algorithm tests the quality of the cubic Hermite interpolant at the real critical points of the interpolant and at nn additional points placed regularly between the interval endpoints. The paper explicitly leaves unresolved whether this fixed regularly spaced choice is optimal, so the problem concerns the selection of test-point locations and their effect on the algorithm's reliability or efficiency.

References

Remark: we don't know if this choice is optimal.

— Root isolation for analytic functions using cubic hermite interpolation  (2610.02934 - Raffalli, 2 Oct 2026) in Definition of $(\varepsilon,n)$-good intervals, Section "Hermite interpolation predicate"