Barycentric Lagrange Interpolation As discussed by Jean-Paul Berrut and Lloyd N. Trefethen (2004)

This text discusses barycentric Lagrange interpolation based on the SIAM REVIEW article of Jean-Paul Berrut and Lloyd N. Trefethen [1]. It also offers additional background information, as well as some MATLAB demonstrations. Interpolation Given a set Dn of n + 1 nodes x j with corresponding values f j where j = 0, . . . ,n, we aim to construct the polynomial that satisfies p(x j) = f j j = 0, . . . ,n i.e. the polynomial interpolates Dn. Note that the f j do not necessarily have to correspond to a function. Theorem 1. There exists a unique polynomial pn(x) = p01...n(x) of degree less than or equal to n interpolating Dn [3] There are many different kinds of interpolation; here we focus on Lagrange interpolation. Lagrange interpolation This data set can be interpolated by the Lagrange form of the interpolation polynomial [3], p01...n(x) = n ∑ j=0 l j(x) f (x j), where (0.1) l j(x) = ∏k=0,k 6= j(x− xk) ∏k=0,k 6= j(x j− xk) , (0.2) also called the Lagrangian cardinal functions [3]. These satisfy, ∀i, j = 0,1 . . .n: li(x j) = δi j = { 0 i f i 6= j 1 i f i = j p01...n(xi) = fi