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

  • Authors:

  • Folkmar Bornemann

  • Affiliations:

  • Technische Universität München, Zentrum Mathematik—M3, 80290, München, Germany

  • Venue:
  • Foundations of Computational Mathematics
  • Year:
  • 2011

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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.