Polynomial Interpolation Error

Interpolation Error at a point $\xi$

$$\boxed{r(x)=-\frac{w(x)}{(N+1)!}{f}^{(N+1)}(\xi (x^{\ast }))}$$

Where,

• Error: $r(x)=p(x)-f(x)$
• Newton Polynomial: $w(x)={\prod }_{i=0}^N(x-x_i)$
• Order of Polynomial → $N$
• Number of points → $N+1$
• $f^{(N+1)}(\xi (x^{\ast }))$ → Derivative of function $f$ at point $\xi$ in the domain.

Maximum Interpolation error

$$|r(x)|\le \frac{|\omega (x)|}{(N+1)!}\cdot M_{N+1}\text{ with }{M}_{N+1}=\underset{x\in [a,b]}{\max }\left|f^{(N+1)}(x)\right|.$$

In general, this estimation of $|r(x)|$ does not get better with increasing $N$.

Solution Method