Fast and accurate computation of divided differences for analytic functions, with an application to the exponential function

In this work we analyze in depth the structure of the matrix representing the operator mapping the coefficients of a polynomial p(x) in the monomial basis M into those in the Newton basis N. A factorization in minimal terms of said matrix has been obtained and as a consequence a factorization of its inverse is available too. As a result, a new high performances routine for the computation of the divided differences of the exponential and the closely related phi(l) functions has been produced with satisfying results.