Lagrange interpolation polynomial

The interpolating polynomial is

For distinct interpolation nodes $x_0,\dots ,x_n$, the polynomial ${\ell }_i(x)={\prod }_{j\ne i}(x-x_j)/(x_i-x_j)$ has ${\ell }_i(x_j)={\delta }_{ij}$. These cardinal polynomials form a basis of the polynomials of degree at most $n$, and give $p(x)={\sum }_{i}p(x_i){\ell }_i(x)$.