The Lagrange Interpolating Polynomial is a simple-to-understand but mathematically clunky way to construct an -order polynomial interpolant to a set of data points .
The best way to see it is by example for a quadratic fit to three data points . Let be
It's easy to see the polynomial goes through each data point. If you plug in , the second and third terms vanish, and the fraction in the first term is , so that . Similar simplifications occur when plugging in , to get , and similar for . Also, since everything on the right-hand-side except is a constant, it's clear that the is a polynomial in of order 2. The generalization to higher-order polynomials is straightforward.
For example, let's fit a 2nd order polynomial to . Plugging into the formula gives
This is mathematically clunky because it expresses a quadratic polynomial as the sum of three quadratics, and we have a lot of tedious algebra to do to simplify. If we do that, we get
which can be easily verified as passing through the given data points.