Polynomial Interpolation - Interpolation Error

Interpolation Error

When interpolating a given function f by a polynomial of degree n at the nodes x₀,...,x_n we get the error

where

is the notation for divided differences.

If f is n + 1 times continuously differentiable on a closed interval I and be a polynomial of degree at most n that interpolates f at n + 1 distinct points {x_i} (i=0,1,...,n) in that interval. Then for each x in the interval there exists in that interval such that