How to overcome the numerical instability of the scheme of divided differences?

Abstract

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 Björck 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.