About the Cover: Complex Finite Differences of Higher Order

Abstract

In his recent work, Bengt Fornberg describes the construction of finite difference schemes (FDS) for accurate numerical computation of higher order derivatives of analytic functions. In this note we introduce the characteristic function of these schemes and explore how it encodes properties of the FDS. Visualizations of the characteristic function and their modifications allow one to read off these properties by visual inspection of phase portraits. The cover of this volume shows a phase portrait of a function which is related to a FDS with nine nodes that approximates the 4th derivative with an error of order \(h^8\).