Lagrange Polynomial Interpolation

The goal of Lagrange interpolation is to find the polynomial $p \in P_n$ for $n+1$ interpolation points $x_0, \dots, x_n \in \R$ and associated values $y_0, \dots, y_n \in \R$ so that: $p(x_i) = y_i \quad{} \text{for } i=0, \dots , n$. For the constuction, Lagrange basis functions are used: $L^{(n)}_i(x) := \prod_{\substack{j=0, j \neq i}}^n \frac{x - x_j}{x_i - x_j} \in P_n, i = 0, \dots{} n$. With this the interpolation polynomial can be constructed: $p := \sum^{n}_{i=0} y_i L^{(n)}_i \in P_n$.

References:

• Lagrange polynomial