Finite‐difference approximation for the u(k)‐derivative with O(hM−k+1) accuracy: An analytical expression

Abstract

AbstractAn approximation of function u(x) as a Taylor series expansion about a point x0 at M points xi, ∼ i = 1,2,…,M is used where xi are arbitrary‐spaced. This approximation is a linear system for the derivatives u(k) with an arbitrary accuracy. An analytical expression for the inverse matrix A−1 where A = [Aik] = ${1\over k!}$(xi − x0)k is found. A finite‐difference approximation of derivatives u(k) of a given function u(x) at point x0 is derived in terms of the values u(xi). © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006