# Interpolation

The most important type of interpolation is polynomial interpolation. Given a set of ${\displaystyle n+1}$ data points ${\displaystyle (x_{i},y_{i})}$ (for i=0..n), where no two ${\displaystyle x_{i}}$ are the same, one is looking for a polynomial ${\displaystyle P(x)}$ of degree at most ${\displaystyle n}$, satisfying ${\displaystyle P(x_{i})=y_{i}}$ for i=0..n. It is known, that such a polynomial always exists and is unique.
${\displaystyle P(x)=\sum _{i=0}^{n}y_{i}\prod _{j=0,j\neq i}^{n}{\frac {x-x_{j}}{x_{i}-x_{j}}}}$