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Foundations of Computational Mathematics

, Volume 11, Issue 1, pp 1–63 | Cite as

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

  • Folkmar Bornemann
Article

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 nontrivial applications are discussed in detail.

Keywords

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

Mathematics Subject Classification (2000)

65E05 65D25 65G56 30D15 

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Copyright information

© SFoCM 2010

Authors and Affiliations

  1. 1.Zentrum Mathematik—M3Technische Universität MünchenMünchenGermany

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