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

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 c...

• Type: Journal
• Year: 2011
• Where: FOCM
• Authors: Folkmar Bornemann
• Keywords: Analytic Functions, Entire Functions, FOCM 2011, Regular Growth Mathematics