Finite-difference approximation for the u^(k)-derivative with O(h^M-k+1) accuracy: An analytical expression

An approximation of function u(x) as a Taylor series expansion about a point x₀ at M points x_i, ∼i = 1,2,..., M is used where λ_i, are arbitrary-spaced. This approximation is a linear system for the derivatives u^(λ) with an arbitrary accuracy. An analytical expression for the inverse matrix A^-1 where A = [A_ik] = 1/k^!(x_i - 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 w(x_i).