OP-00231 Solution claimed

A uniform disk bound for wave-kernel Padé approximants

Note: OpenProblemsInNLA (MF-03) lists this problem as solved in Lean, by the theorem NLA.Proofs.MF03.all_order_target at https://github.com/ajt60gaibb/OpenProblemsInNLA/tree/5487945e5a72abd952b0c89915915de8cc9b3e65/lean-statements/NLA/Proofs/MF03/AllOrderTarget.lean. It is part of OpenProblemsInNLA's shared Lean library rather than a standalone project, which the registry's verifier does not record yet, so it has not been checked here.

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Let $$f(z) = \cosh\sqrt z = \sum_{j=0}^{\infty} \frac{z^j}{(2j)!},$$ an entire function. For an integer $m \ge 1$ let $r_m = P_m/Q_m$ be its diagonal Padé approximant at $0$: $\deg P_m, \deg Q_m \le m$, $Q_m(0) = 1$ and $Q_m(z) f(z) - P_m(z) = O(z^{2m+1})$, with $r_m$ taken as a reduced rational function. Nadukandi and Higham base an algorithm for the wave-kernel matrix functions $\cosh\sqrt A$ and $\operatorname{sinhc}\sqrt A$, where $\operatorname{sinhc} z = \sinh z/z$, on these approximants. In their error analysis, the bound below lets the condition $\rho(A) \le 3$ on the spectral radius replace $\rho(I - r_m(A)) \le 2$. They proved, from the coefficients of $P_m$ and $Q_m$ computed symbolically, that for $1 \le m \le 20$ the approximant $r_m$ has no pole in the disk $|z| \le 3$ and satisfies the bound there.

Problem. Is it true that, for every $m \ge 1$, $r_m$ has no pole in $\{z \in \mathbb{C} : |z| \le 3\}$ and $$|1 - r_m(z)| \le 2 \qquad (|z| \le 3)?$$

Source: Nadukandi and Higham (2018), Conjecture 4.6.

Imported from github.com/ajt60gaibb/OpenProblemsInNLA.

Classification

Primary: NA — Numerical Analysis
Secondary: CA
Keywords: backward error analysis, matrix functions, OpenProblemsInNLA, Padé approximants
Literature checked: Oct. 8, 2026 by clemens-admin

References

@article{nadukandi2018,
  author  = {Nadukandi, Prashanth and Higham, Nicholas J.},
  title   = {Computing the Wave-Kernel Matrix Functions},
  journal = {SIAM J. Sci. Comput.},
  volume  = {40},
  number  = {6},
  year    = {2018},
  pages   = {A4060--A4082},
  doi     = {10.1137/18M1170352},
}
@misc{openproblemsinnla-mf03,
  author       = {Townsend, Alex},
  title        = {{MF-03}: A uniform disk bound for wave-kernel Padé approximants},
  howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
  year         = {2026},
  url          = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-functions-and-stability/MF-03/README.md},
  note         = {Read at commit eb7c9d6, 2026-10-07},
}

Resolution claims

A uniform disk bound for diagonal Padé approximants of the wave-kernel function v1 GitHub ↗

Matthew J. Colbrook

Let $r_m$ be the diagonal $[m/m]$ Padé approximant at the origin to $\cosh\sqrt z$. We prove that $r_m$ exists, has no poles in $|z| \le 3$, and satisfies $|r_m(z) - 1| \le 2$ there, for every $m \ge 0$. The bound is strict for $m \ge 2$ and is attained for $m = 1$ at $z = 3$. The large-order …

Full resolution · not verified · fidelity not established · posted on GitHub 11 Sep 2026

Status history

When	Change	By	Reason
2026-10-08	Needs verification → Solution claimed	clemens-admin	Listed with the record of its solution, unreviewed.

Citing this problem

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OP-00231, "A uniform disk bound for wave-kernel Padé approximants", clopen.solutions.