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

TL;DR

This paper investigates the numerical stability of computing high-order derivatives of analytic functions via Cauchy integrals, identifying an optimal radius that minimizes round-off errors and achieves near-full accuracy for many functions.

Contribution

It provides a comprehensive analysis of the stability and accuracy of Cauchy integral methods for derivatives, including the identification of a unique optimal radius and theoretical explanations.

Findings

Existence of a unique optimal radius for minimal round-off error.

Optimal radius often yields near-full accuracy for many classes of functions.

Theoretical explanations using Hardy spaces, Wiman-Valiron, Levin-Pfluger theory, and saddle-point method.

Abstract

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.