  The scheme of divided differences is widely used in many approximation and
interpolation problems. Computing the Newton coefficients of the interpolating
polynomial is the first step of the Bj\"{o}rck and Pereyra algorithm for
solving Vandermonde systems of equations (Cf. \cite{bjorck: 70}). Very often
this algorithm produces very accurate solution. The problem of determining the
Newton coefficients is intimately related with the problem of evaluation the
Lagrange interpolating polynomial, which can be realized by many algorithms.
For these reasons we use the uniform approach and analyze also Aitken's
algorithm of the evaluation of an interpolating polynomial. We propose new
algorithms that are always numerically stable with respect to perturbation in
the function values and more accurate than the Aitken's algorithm and the
scheme of divided differences, even for complex data.
