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

• Published in: Foundations of computational mathematics Vol. 11; no. 1; pp. 1 - 63
• Main Author: Bornemann, Folkmar
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.02.2011; Springer; Springer Nature B.V
• Subjects: Accuracy; Analytic functions; Applications of Mathematics; Asymptotic properties; Computation; Computer Science; Derivatives; Economics; Integral equations; Integrals; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Entire functions of perfectly and completely regular growth; Accuracy; Stability; Analytic functions; Hardy spaces; Optimal radius; 65E05; 65D25; 65G56; Cauchy integral; Numerical differentiation; 30D15
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-010-9075-z

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.