Accuracy and Stability of Computing High-order Derivatives of Analytic Functions by Cauchy Integrals

作者：Bornemann Folkmar^*

来源：Foundations of Computational Mathematics, 2011, 11(1): 1-63.

DOI：10.1007/s10208-010-9075-z

• Numerical differentiation
• Accuracy
• Stability
• Analytic functions
• Cauchy integral
• Optimal radius
• Hardy spaces
• Entire functions of perfectly and completely regular growth

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.

• 出版日期2011-2