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arXiv · 0910.1841

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

Folkmar Bornemann
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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 non-trivial applications are discussed in detail.

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Folkmar Bornemann. 2009-10-09. Accuracy and Stability of Computing High-Order Derivatives of Analytic Functions by Cauchy Integrals. https://doi.org/10.1007/s10208-010-9075-z

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