# Lagrange Form (Interpolation)

The lagrange basis is a different basis for interpolating polynomials.

We will define the Lagrange basis functions, $L_{i}(x)$, to construct a polynomial as
$p(x)=y_{1}L_{1}(x)+y_{2}L_{2}(x)+⋯+y_{n}L_{n}(x)=∑_{i=1}y_{i}L_{i}(x)$
where $y_{i}$ are the coefficients (**and** are also our data values, $y_{i}=p(x_{i}))$.
$L_{i}(x)=(x_{i}−x_{1})...(x_{i}−x_{i−1})(x_{i}−x_{i+1})...(x_{i}−x_{n})(x−x_{1})...(x−x_{i−1})(x−x_{i+1})...(x−x_{n}) $

Why use Lagrange form over a Vandermonde System?

One advantage of the Lagrange form is that the interpolating polynomial can be written down directly, without needing to solve a (Vandermonde) linear system.