High Degree Polynomial Interpolation in Newton Form

Polynomial interpolation is an essential subject in numerical analysis. Dealing with a real interval, it is well known that even if $f(x)$ is an analytic function, interpolating at equally spaced points can diverge [P. J. Davis, Interpolation and Approximation, Dover, New York, 1975]. On the other hand, interpolating at the zeros of the corresponding Chebyshev polynomial will converge. Using the Newton ,formula, this result of convergence is true only on the theoretical level. It is shown that the algorithm which computes the divided differences is numerically stable only if: (1) The interpolating points are arranged in a certain order. (2) The size of the interval is four. Stabilizing Newton interpolation is described in greater generality of interpolation in the complex plane.High degree polynomial interpolation is employed in approximation of functions of matrices. Some applications of the results described in the paper for this purpose are given.