On a positivity conjecture of Fasi and Higham for Padé approximants to the matrix exponential
Abstract
We study the positivity conjecture of Fasi and Higham [SIAM J. Matrix Anal. Appl. 40 (2019), Conjecture 3.2], which concerns the Taylor coefficients of the reciprocal of the denominator qk,m(z) of the Padé approximant to the matrix exponential. The conjecture asserts that these coefficients are positive whenever m is odd. We prove full positivity for k=0 and eventual positivity whenever the unique positive real zero of qk,m is the unique zero of minimal modulus. We also give an explicit computable sufficient threshold beyond which all coefficients are positive. Using Hermite scaling with leading-coefficient normalization, we prove that this spectral minimality holds for each fixed odd m and all sufficiently large k. The unrestricted conjecture is nevertheless false. We give exact rational counterexamples and prove a transient scaling theorem showing that on the scale nasymp√(k+1) the normalized coefficients converge to an explicit oscillatory Hermite profile. For m=5 this profile is negative at one point, yielding infinitely many counterexamples. The counterexamples are not diagonal, and the diagonal case k=m, which is the case relevant in practice, remains open. These results do not imply a failure of the Fasi--Higham algorithm. Rather, they delimit the range in which the auxiliary positivity conjecture can be used as a theorem.
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BibTeX
@misc{scardino2026positivity,
title = {{On a positivity conjecture of Fasi and Higham for Padé approximants to the matrix exponential}},
author = {Francesco Scardino},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39051},
eprint = {2610.39051},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39051},
}
RIS
TY - UNPB
TI - On a positivity conjecture of Fasi and Higham for Padé approximants to the matrix exponential
AU - Francesco Scardino
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.39051
UR - https://arxiv.org/abs/2610.39051
AB - We study the positivity conjecture of Fasi and Higham [SIAM J. Matrix Anal. Appl. 40 (2019), Conjecture 3.2], which concerns the Taylor coefficients of the reciprocal of the denominator q_k,m(z) of the Padé approximant to the matrix exponential. The conjecture asserts that these coefficients are positive whenever m is odd. We prove full positivity for k=0 and eventual positivity whenever the unique positive real zero of q_k,m is the unique zero of minimal modulus. We also give an explicit computable sufficient threshold beyond which all coefficients are positive. Using Hermite scaling with leading-coefficient normalization, we prove that this spectral minimality holds for each fixed odd m and all sufficiently large k. The unrestricted conjecture is nevertheless false. We give exact rational counterexamples and prove a transient scaling theorem showing that on the scale nasymp√(k+1) the normalized coefficients converge to an explicit oscillatory Hermite profile. For m=5 this profile is negative at one point, yielding infinitely many counterexamples. The counterexamples are not diagonal, and the diagonal case k=m, which is the case relevant in practice, remains open. These results do not imply a failure of the Fasi--Higham algorithm. Rather, they delimit the range in which the auxiliary positivity conjecture can be used as a theorem.
ER -