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,: \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.

More information on Lagrange Basis Polynomials is here

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

so expanding this,: \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.