具有实极点的矩阵指数的时间一致有理逼近
Uniform-in-time rational approximation of the matrix exponential with real poles
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中文总结 AI 辅助
提出两种新方法构造有理函数族逼近\(\exp(-tz)\)，一种针对实极点合并情况，通过权重函数得出极点最优位置公式；另一种针对实极点不同情况，利用佐洛塔廖夫构造给出算法，还分析了稳定性，得到可靠且可并行化的指数传播子。
中文摘要 AI 辅助
我们提出了两种新方法来构造具有共享实极点的有理函数族，这些函数族在正时间区间内对\(z\geq0\)和\(t\)的函数\(\exp(-tz)\)进行几乎一致逼近。第一个结果涉及所有实极点合并为一个点的情况。通过适当选择权重函数，我们能够推导出这样一个极点的渐近最优位置的封闭公式。然后我们讨论所有实极点都不同的更一般情况。利用佐洛塔廖夫对某些最优有理函数的构造，我们提出了一种简单算法来有效地推导几乎最优的极点。我们分析了所得有理矩阵函数在浮点运算中的数值评估稳定性。通过控制由部分分式产生的潜在病态增长，获得了可靠且高度可并行化的指数传播子。
英文摘要
We propose two new approaches for constructing families of rational functions with shared real poles that nearly uniformly approximate the functions $\exp(-tz)$ for $z\geq 0$ and $t$ in a positive time interval. The first result concerns the case where all real poles coalesce into a single point. With an appropriate choice of a weight function we are able to derive a closed formula for the asymptotically optimal location of such a pole. We then discuss the more general case where all real poles are distinct. Using Zolotarev's construction of certain optimal rational functions, we present a simple algorithm to derive nearly optimal poles efficiently. We analyze the stability of the numerical evaluation of the resulting rational matrix functions in floating-point arithmetic. By controlling the growth of potential ill-conditioning arising from partial fractions, reliable and highly parallelizable exponential propagators are obtained.
