Accuracy and Stability of Computing High-order Derivatives of Analytic Functions by Cauchy Integrals
Accuracy and Stability of Computing High-order Derivatives of Analytic Functions by Cauchy Integrals
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DOI：
10.1007/s10208-010-9075-z
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发表时间：
2011-02-01
影响因子：
3
通讯作者：
Bornemann, Folkmar
中科院分区：
文献类型：
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作者：
Bornemann, Folkmar
解析函数的高阶导数可表示为圆形轮廓上的柯西积分，这可以非常有效地近似，例如，通过梯形和。然而在解析上，直到收敛半径的每个半径r都是相等的，数值稳定性强烈地依赖于r。我们对这种影响进行了全面的研究；特别是，我们证明了存在一个唯一的半径，它可以使舍入误差造成的精度损失最小化。对于大的函数类，虽然不是所有的，这个半径实际上给出了完全的精度；我们用Hardy空间理论、全函数的Wiman-Valiron理论和Levin-Pfluger理论以及渐近分析的鞍点方法来解释这个显著的事实。详细讨论了许多示例和重要应用。
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 nontrivial applications are discussed in detail.