General explicit difference formulas for numerical differentiation

Through introducing the generalized Vandermonde determinant, the linear algebraic system of a kind of Vandermonde equations is solved analytically by use of the basic properties of this determinant, and then we present general explicit finite difference formulas with arbitrary order accuracy for approximating first and higher derivatives, which are applicable to unequally or equally spaced data. Comparing with other finite difference formulas, the new explicit difference formulas have some important advantages. Basic computer algorithms for the new formulas are given, and numerical results show that the new explicit difference formulas are quite effective for estimating first and higher derivatives of equally and unequally spaced data.