# Polynomial Interpolation#

For a given set of $$k + 1$$ points $$\left(x_{j}, y_{j}\right)$$ with no two $$x_{j}$$ values equal, to find the polynomial of lowest degree that assumes at each value $$x_{j}$$ the corresponding value $$y_{j}$$.

## Lagrange interpolation polynomial#

$L(x):=\sum_{j=0}^{k} y_{j} \ell_{j}(x)$

### Lagrange basis polynomials:#

$\begin{split} \ell_{j}(x):=\prod_{\substack{0 \leq m \leq k \\ m \neq j}} \frac{x-x_{m}}{x_{j}-x_{m}} \quad ,j\in [0,k] \end{split}$

Note that:

$\forall(i \neq j): \ell_{j}\left(x_{i}\right)=0$
$\ell_{j}\left(x_{j}\right):=1$

It follows that $$y_{j} \ell_{j}\left(x_{j}\right)=y_{j}$$, $$L\left(x_{j}\right)=y_{j}$$