03 - Error Formula for Polynomial Interpolation

Error Formula for Polynomial Interpolation

Let $f$ be $(n+1)$-times continuously differentiable on interval
Let $p_n$ be the polynomial interpolant at $\{x_i{\}}_{i=0}^n$

For every $x\in [x_0,x_n]$ there exists $\xi =\xi (x)\in (x_0,x_n)$ such that

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

where $f^{(n+1)}$ is the $(n+1)$st derivative of $f$

Proof

For $x=x_k$ where $k=0,1,\cdots ,n$ then

$$e(x)=0$$

by definition

Otherwise suppose $x\ne x_k$

Let

$$\varphi (t)\equiv e(t)-\frac{e(x)}{\pi (x)}\pi (t)$$

where

$$\begin{array}{rl}\pi (t)& \equiv (t-x_0)(t-x_1)\cdots (t-x_n)\\ & =t^{n+1}-\left(\sum _{i=0}^n{x}_i\right)t^n+\cdots +(-1{)}^{n+1}{x}_0{x}_1\cdots x_n\\ & \in {\Pi }_{n+1}\end{array}$$

As

$$\varphi (t)=0\text{ at }(n+2)\text{ points }\text{ where }t=x_0\text{ and }t=x_k\text{ for }k=0,1,\cdots ,n$$

Then

$${\varphi }^{\prime }(t)=0\text{ at }(n+1)\text{ points }\text{ for }t={\xi }_k\text{ for }k=0,\cdots ,n$$

Then

$${\varphi }^{\prime \prime }(t)=0\text{ at }n\text{ points }\text{ for }t={\overline{\xi }}_k\text{ for }k=0,\cdots ,n-1$$

Hence

$$\exists \xi \in (x_0,x_n)\text{ such that }{\varphi }^{(n+1)}(\xi )=0$$

However

$${\varphi }^{(n+1)}(t)=e^{(n+1)}(t)-\frac{e(x)}{\pi (x)}{\pi }^{(n+1)}(t)=f^{(n+1)}(t)-\frac{e(x)}{\pi (x)}(n+1)!$$

As $p_n^{(n+1)}(t)\equiv 0$ and $\pi (t)$ is a monic polynomial of order $(n+1)$ then result follows