05 - Error Formula for Hermite Interpolating Polynomials

Error Formula for Hermite Interpolating Polynomials

Let $f$ be a function with at least $2n+2$ smooth derivatives

Let $p_{2n+1}$ be the Hermite Interpolating Polynomial where

$$f_i=f(x_i)\text{ and }{g}_i=f^{\prime }(x_i)$$

Then for all $x\in [x_0,x_n]$, there exists $\xi \in (x_0,x_n)$ such that

$$e(x)\equiv f(x)-p_{2n+1}(x)=[(x-x_0)(x-x_1)\cdots (x-x_n){]}^2\frac{f^{(2n+2)}(\xi )}{(2n+2)!}$$

where $f^{(2n+2)}$ is the $(2n+2)$th derivative of $f$