Lagrange Polynomials

Lagrange polynomials provide a way to construct a degree \(n\) polynomial from \(n+1\) available points \((x_i, f(x_i))\).

For each point \(k\), the basis polynomial \(L_{n,k}\) is defined as:

\[ L_{n,k}(x) = \prod\limits_{\substack{i=0 \\ i \neq k}}^n\frac{{(x-x_i)}}{{(x_k-x_i)}} \]

Analogy

It can be seen as a fraction where the numerator is the product of all \((x - x_i)\) terms except for \(i = k\), and the denominator is the same product but evaluated at \(x = x_k\).

The Interpolaing Polynomial

Finally, the \(n\)-degree polynomial is constructed as a linear combination of the function values and their respective basis polynomials:

\[ P_n(x) = \sum_{i = 0}^n f(x_i)L_{n,i}(x) \]

This polynomial guarantees that \(P_n(x_i) = f(x_i)\) for all given nodes.