Lagrange basis polynomial

One of n+1 building-block polynomials, each equal to 1 at its own node and 0 at every other node -- combining them with the data values assembles the interpolating polynomial directly.

For nodes $x_0,\ldots,x_n$ , the $i$ -th Lagrange basis polynomial is $L_i(x)=\prod_{j\ne i}\frac{x-x_j}{x_i-x_j},$ satisfying $L_i(x_i)=1$ and $L_i(x_j)=0$ for $j\ne i$ . The interpolating polynomial is then $P(x)=\sum_{i=0}^n y_i L_i(x)$ -- each term contributes exactly $y_i$ at its own node and nothing at any other, so the sum matches every data point by construction.

Examples

For nodes $0,1,2$ : $L_0(x)=\dfrac{(x-1)(x-2)}{(0-1)(0-2)}$ , which equals $1$ at $x=0$ and $0$ at both $x=1$ and $x=2$ .