Lagrange Interpolating Polynomials

Definition

given $(N+1)$ unique data points
* $(x_0,y_0),(x_1,y_1),....,(x_N,y_N)$
we can create an $N^{th}$ order Lagrange interpolating polynomial

\[P_n(x) = \sum_{i=0}^N \mathcal{L}_i(x)f(x_i)\]

where, : $:nowrap:$$$$\begin{eqnarray} f(x_0) = y_0\\ f(x_1) = y_1\\ .\\ .\\ f(x_i) = y_i\\ .\\ f(x_N) = y_N \end{eqnarray}$

So, we are just multiplying by the given $y_i$ values.

Lagrange Basis Polynomials

More information on Lagrange Basis Polynomials is here

\[\mathcal{L}_i(x)=\prod_{\substack{j=0 \\ j\neq i}}^{N}\frac{x-x_j}{x_i-x_j}\]

so expanding this, : $:nowrap:$$$$\begin{eqnarray} \mathcal{L}_i(x) &=\frac{x-x_0}{x_i-x_0}\frac{x-x_1}{x_i-x_1}...\\ &...\frac{x-x_{i-1}}{x_i-x_{i-1}}...\\ &...\frac{x-x_{i+1}}{x_i-x_{i+1}}...\\ &...\frac{x-x_N}{x_i-x_N} \end{eqnarray}$

Notice that we do not include the term where $i==j$!

Please see lpf for details on implementation.