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 non-trivial applications are discussed in detail.

Version 4 has some references and a discussion of other quadrature rules added; 57 pages, 7 figures, 6 tables; to appear in Found. Comput. Math

References in corpus (1)

• On the Numerical Evaluation of Fredholm Determinants

Cited by in corpus (7)

• From Painlevé to Zakharov-Shabat and beyond: Fredholm determinants and integro-differential hierarchies
• Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles
• Construction and implementation of asymptotic expansions for Jacobi--type orthogonal polynomials
• A Stirling-type formula for the distribution of the length of longest increasing subsequences
• Asymptotic expansions relating to the distribution of the length of longest increasing subsequences
• Numerical initial data deformation exploiting a gluing construction: I. Exterior asymptotic Schwarzschild
• On the Conjecture of Stability Preservation in Arbitrary-Order Adams-Bashforth-Type Integrators