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.
Contour-integral based quantum eigenvalue transformation: analysis and applications
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）
Contour Hankel dynamics and indicator fields for the Riemann $Ξ$-function
School of Mathematical Science, University of Electronic Science and Technology of China（电子科技大学数学科学学院）；School of Mathematics, Southwest Jiaotong University（西南交通大学数学系）
Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivatives
Hong Kong University of Science and Technology（香港科技大学）；University of Maryland College Park（马里兰大学帕克分校）；HKUST Shenzhen-Hong Kong Collaborative Innovation Research Institute（港科大深港协同创新研究院）
