Optimal Contours for High-Order Derivatives

TL;DR

This paper develops a method to find optimal contour paths for calculating high-order derivatives of holomorphic functions, improving numerical stability especially for functions with singularities.

Contribution

It introduces a novel approach using Provan's algorithm to minimize the condition number over grid paths, extending beyond circular contours.

Findings

Optimal rectangular paths reduce condition numbers effectively.

Method outperforms circular contours in functions with branch cuts.

Numerical examples validate the approach's robustness.

Abstract

As a model of more general contour integration problems we consider the numerical calculation of high-order derivatives of holomorphic functions using Cauchy's integral formula. Bornemann (2011) showed that the condition number of the Cauchy integral strongly depends on the chosen contour and solved the problem of minimizing the condition number for circular contours. In this paper we minimize the condition number within the class of grid paths of step size h using Provan's algorithm for finding a shortest enclosing walk in weighted graphs embedded in the plane. Numerical examples show that optimal rectangular paths yield small condition numbers even in those cases where circular contours are known to be of limited use, such as for functions with branch-cut singularities.