About the cover: complex finite differences of higher order (Q6200995)

To compute the \(k\)th derivative \(f^{(k)}(0)\) of a function \(f\) analytic in a neighborhood of the origin, \textit{B. Fornberg} [IMA J. Numer. Anal. 41, No. 2, 814--825 (2021; Zbl 1518.65024); Numer. Algorithms 90, No. 3, 1305--1326 (2022; Zbl 1492.65063)] started with nodes \(z_1, \ldots, z_N\) and constructed (complex) weights \(w_1, \ldots, w_N\) such that \[ f^{(k)}(0)-\frac{1}{h^k}\sum_{j=1}^N w_j f(hz_j)=O(h^m) \] as \(h \to 0\). If \(m\) is the largest natural number such that this is true for all functions \(f\) analytic in a neighborhood of the origin, one calls the family of nodes and weights a stencil of differentiation order \(k\) and approximation order \(m\). The theorem established by the author in this paper says that a family of nodes and weights forms a stencil of differentiation order \(k\) and approximation order \(m\) if and only if there is a constant \(c \neq 0\) such that \[ w(z):=\sum_{j=1}^N \frac{w_j}{z-z_j}=\frac{k!}{z^{k+1}}+\frac{c}{z^{k+m+1}} +O\left(\frac{1}{z^{k+m+2}}\right) \] as \(z \to \infty\). This theorem allows one to compute the best weights for arbitrary nodes by solving a linear system. Moreover, the author shows how the differentiation order \(k\) and the approximation order \(m\) of a family of nodes and weights can be read off by visual inspection of the phase portraits of the function \(w(z)\) and its modifications.