The formulae and algorithms for Lagrange-power basis transformation and Lagrange–Newton transformation

The interpolation polynomials based on Lagrange, Newton and power basis play important roles in applied mathematics, computing method and many other emerging applications. In this paper, we present some coordinate transformation formulae and algorithms as demonstrated below. Firstly, we put forward the formulae of the Lagrange-power basis transformation and its inverse transformation, and as a byproduct, we provide a new method to arrive at the inversion of the Vandermonde matrix. Secondly, we give the formulae of Lagrange–Newton transformation and its inverse transformation. Moreover, we construct related algorithms of Lagrange-power basis transformation, Lagrange–Newton transformation and their inverse transformations.