Optimal Contours for High-Order Derivatives

Details Optimal Contours for High-Order Derivatives Optimal Contours for High-Order Derivatives

2011-07-03

arxiv.org/abs/1107.0498v2

Mathematics

Numerical Analysis

added discussion of using diagonals in the grid, added figure illustrating spectral accuracy; 9 pages, 5 figures

Scientific paper

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 rectangular 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 not much use, such as for functions with branch-cut singularities.

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