Lagrange Interpolating Polynomials

A rearranged form of Newton’s divided difference polynomial.

The $k$-th Lagrange basis polynomial for $n+1$ nodes is:

Here:

• $x_0, x_1, \ldots, x_n$: the $n+1$ distinct data points (nodes)
• $L_{n,k}(x_k) = 1$ and $L_{n,k}(x_i) = 0$ for $i \neq k$

The $n$-th Lagrange interpolating polynomial through the data $(x_k, f(x_k))$ is:

Example

Interpolate through $(1, 1)$, $(2, 4)$, $(3, 9)$ (i.e., $f(x) = x^2$).

Basis polynomials for $n = 2$:

At $x = 4$: $L_{2,0}(4) = 1$, $L_{2,1}(4) = -3$, $L_{2,2}(4) = 3$, so:

$P_2(4) = 1(1) + 4(-3) + 9(3) = 1 - 12 + 27 = 16 = 4^2 \checkmark$