Optimal contours for high-order derivatives

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, Accuracy and stability of computing high-order derivatives of analytic functions by Cauchy integrals. Found. Comput. Math., 11, 1-63) 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 grid 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.