通过柯西积分计算解析函数高阶导数的准确性与稳定性
Accuracy and Stability of Computing High-Order Derivatives of Analytic Functions by Cauchy Integrals
AI总结：
本文研究通过柯西积分计算解析函数高阶导数时，如何选择最优半径以平衡截断误差和舍入误差，并利用Hardy空间、Wiman-Valiron和Levin-Pfluger整函数理论以及鞍点法解释为何该半径通常能实现几乎全精度。
摘要：
解析函数的高阶导数可以表示为沿圆形轮廓的柯西积分，这可以通过梯形求和等方法非常有效地近似。尽管在解析上，直到收敛半径的每个半径r都是等价的，但数值稳定性强烈依赖于r。我们对这种效应进行了全面研究；特别地，我们表明存在一个唯一的半径，可以最小化由舍入误差引起的精度损失。对于大多数函数类（尽管并非所有），该半径实际上能提供几乎全精度；这一显著事实我们通过Hardy空间理论、Wiman-Valiron和Levin-Pfluger整函数理论以及渐近分析的鞍点法进行了解释。详细讨论了多个例子和非平凡应用。
英文摘要：
High-order derivatives of analytic functions are expressible as Cauchy integrals over circular contours, which can very effectively be approximated, e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius of convergence is equal, numerical stability strongly depends on r. We give a comprehensive study of this effect; in particular we show that there is a unique radius that minimizes the loss of accuracy caused by round-off errors. For large classes of functions, though not for all, this radius actually gives about full accuracy; a remarkable fact that we explain by the theory of Hardy spaces, by the Wiman-Valiron and Levin-Pfluger theory of entire functions, and by the saddle-point method of asymptotic analysis. Many examples and non-trivial applications are discussed in detail.
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