Fasi–Higham positivity conjecture for Padé approximants

For integers $k\ge 0$ and odd integers $m\ge 1$ , let $q_{k,m}(z)$ be the denominator, normalized by $q_{k,m}(0)=1$ , of the $[k/m]$ Padé approximant to $e^z$ . Writing $\frac{1}{q_{k,m}(z)}=\sum_{n=0}^{\infty}c_n z^n$ , the conjecture asserts that $c_n>0$ for every $n\ge 0$ .

References

On a positivity conjecture of Fasi and Higham for Padé approximants to the matrix exponential — arXiv — Francesco Scardino

Source: arXiv

matrix-analysis numerical-analysis

Posted 1 day ago · edited 1 day ago

Progress summary

Refreshed 1 day ago

Open · progress reported

A new unrefereed paper gives infinitely many counterexamples to the broad claim, while the important diagonal case remains open.

The conjecture asserts that all Taylor coefficients of the reciprocal Padé denominator are positive for odd values of $m$ . The general claim is challenged by infinitely many counterexamples when $m=5$ , but the diagonal case is not resolved.

October 2026 counterexamples

Francesco Scardino's unrefereed preprint claims infinitely many counterexamples for $m=5$ , disproving the conjecture outside the diagonal case. It identifies a limitation in the positivity premise used to analyze the Fasi–Higham algorithm; the diagonal case remains open.

Current status (as of October 2026): The general positivity conjecture is claimed false through infinitely many counterexamples for $m=5$ , but this claim is unverified and the diagonal case remains open.

Sources

arxiv.org

No solutions have been posted yet.