caa a22 4500 445331488 CHVBK 20180317142739.0 cr unu---uuuuu 170323e20110201xx s 000 0 eng 10.1007/s10208-010-9075-z doi (NATIONALLICENCE)springer-10.1007/s10208-010-9075-z Bornemann Folkmar Zentrum Mathematik—M3, Technische Universität München, 80290, München, Germany aut Accuracy and Stability of Computing High-order Derivatives of Analytic Functions by Cauchy Integrals [Elektronische Daten] [Folkmar Bornemann] 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 onr. 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. SFoCM, 2010 Numerical differentiation nationallicence Accuracy nationallicence Stability nationallicence Analytic functions nationallicence Cauchy integral nationallicence Optimal radius nationallicence Hardy spaces nationallicence Entire functions of perfectly and completely regular growth nationallicence Foundations of Computational Mathematics Springer-Verlag 11/1(2011-02-01), 1-63 1615-3375 11:1<1 2011 11 10208 https://doi.org/10.1007/s10208-010-9075-z text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10208-010-9075-z text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Bornemann Folkmar Zentrum Mathematik—M3, Technische Universität München, 80290, München, Germany aut NATIONALLICENCE 773 0- Foundations of Computational Mathematics Springer-Verlag 11/1(2011-02-01), 1-63 1615-3375 11:1<1 2011 11 10208 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer