How sharp are error bounds? –lower bounds on quadrature worst-case errors for analytic functions
CoRR(2024)
摘要
Numerical integration over the real line for analytic functions is studied.
Our main focus is on the sharpness of the error bounds. We first derive two
general lower estimates for the worst-case integration error, and then apply
these to establish lower bounds for various quadrature rules. These bounds turn
out to be either novel or improve upon existing results, leading to lower
bounds that closely match upper bounds for various formulas. Specifically, for
the suitably truncated trapezoidal rule, we improve upon general lower bounds
on the worst-case error obtained by Sugihara [Numer. Math., 75 (1997),
pp. 379–395] and provide exceptionally sharp lower bounds apart from a
polynomial factor, in particular show that the worst-case error for the
trapezoidal rule by Sugihara is not improvable more than a polynomial factor.
Additionally, our research reveals a discrepancy between the error decay of the
trapezoidal rule and Sugihara's lower bound for general numerical integration
rules, introducing a new open problem. Moreover, Gauss–Hermite quadrature is
proven sub-optimal under the decay conditions on integrands we consider, a
result not deducible from upper-bound arguments alone. Furthermore, to
establish the near-optimality of the suitably scaled Gauss–Legendre and
Clenshaw–Curtis quadratures, we generalize a recent result of Trefethen
[SIAM Rev., 64 (2022), pp. 132–150] for the upper error bounds in
terms of the decay conditions.
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