Finite-difference approximation for the u((k))-Derivative with O(h(M-k+l)) accuracy: an analytical expression

摘要

An approximation of function u(x) as a Taylor series expansion about a point x(o) at M points x(i), similar to i = 1, 2,..., M is used where x(i) 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 = [A(ik)] = 1/k!(x(i) - x(o))(k) is found. A finite-difference approximation of derivatives u((k)) of a given function u(x) at point x(o) is derived in terms of the values u(x(i)). (C) 2006 Wiley Periodicals, Inc.