发表机构
Clemson University; Zhejiang University of Finance and Economics（克莱姆森大学; 浙江财经大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过Möbius变换和双鞍渐近分析，研究了指数函数重极点有理逼近的一致误差，给出了预因子分辨的误差估计和最优极点选择，并应用于矩阵作用基准。
AI 中文摘要
我们研究了在$(-\infty,0]$上由具有给定重极点$q_m>0$的有理函数$P_m(z)/(q_m-z)^m$对$\exp(tz)$（$t>0$）的一致逼近。一个Möbius变换将问题归结为在$[-1,1]$上对$F_\lambda(x)=\exp\\!\left(-\lambda\frac{1-x}{1+x}\right)$的多项式逼近，其中$\lambda=tq_m$。我们推导了切比雪夫系数的统一双鞍渐近公式，包括显式的振幅和相位以及每个局部复鞍贡献的相对余项，覆盖了远离鞍点合并的固定、次线性和线性极点尺度。由于两个鞍点贡献可能在单个系数中相互抵消，我们转而考虑一个增长的相邻系数块，并证明整个块不能抵消。这将系数渐近转化为逼近误差。对于$q_m=(\alpha/t)m$，$0<\alpha<3\sqrt3/2$，最佳一致误差具有双边阶$m^{-1/2}H_e(\alpha)^m$；在最优比率$\alpha=1/\sqrt2$时，这变为$m^{-1/2}(\sqrt2-1)^m$。归一化的切比雪夫加权$L^2$投影误差具有显式的有界振荡轮廓。这些预因子分辨的估计产生了一个两项精度-工作量定律，并量化了极点设计与停止度数之间的不匹配。最后，标量误差为自伴负半定矩阵提供了精确的最坏情况矩阵作用基准，以及维度无关的移位-反转Krylov界。
英文摘要
We study uniform approximation of $\exp(tz)$, $t>0$, on $(-\infty,0]$ by rational functions $P_m(z)/(q_m-z)^m$ with a prescribed repeated pole $q_m>0$. A Möbius transformation reduces the problem to polynomial approximation of $F_λ(x)=\exp\!\left(-λ\frac{1-x}{1+x}\right)$ on $[-1,1]$, where $λ=tq_m$. We derive a uniform two-saddle asymptotic formula for the Chebyshev coefficients, including explicit amplitude and phase and a relative remainder for each localized complex saddle contribution, covering fixed, sublinear, and linear pole scalings away from saddle coalescence. Because the two saddle contributions can cancel in a single coefficient, we pass to a growing block of neighboring coefficients and prove that the whole block cannot cancel. This transfers the coefficient asymptotics to approximation errors. For $q_m=(α/t)m$, $0<α<3\sqrt3/2$, the best uniform error has two-sided order $m^{-1/2}H_e(α)^m$; at the optimal ratio $α=1/\sqrt2$ this becomes $m^{-1/2}(\sqrt2-1)^m$. The normalized Chebyshev-weighted $L^2$ projection error has an explicit bounded oscillatory profile. These prefactor-resolved estimates yield a two-term precision-to-work law and quantify mismatch between pole-design and stopping degrees. Finally, the scalar error gives an exact worst-case matrix-action benchmark for self-adjoint negative semidefinite matrices and dimension-independent shift-and-invert Krylov bounds.
