  This paper provides a new approach to derive various arbitrary high order
finite difference formulae for the numerical differentiation of analytic
functions. In this approach, various first and second order formulae for the
numerical approximation of analytic functions are given with error terms
explicitly expanded as Taylor series of the analytic function. These lower
order approximations are successively improved by one or two (two order
improvement for centered formulae) to give finite difference formulae of
arbitrary high order. The new approach allows to recover the standard backward,
forward, and centered finite difference formulae which are given in terms of
formal power series of finite difference operators. Examples of new formulae
suited for deferred correction methods are given.
