A Free Cubic Spline is also called a Natural Cubic Spline.

First, let's define a set of data with x-coordinates and corresponding y-values [tex]y_0,\dots,y_n[tex]. To determine cubic splines we must have a closed system. To close the system we could choose various conditions. For a natural spline we choose the boundary conditions . For your example this implies . Let's define the cubic spline on the i-th interval by

.

Then we can define a system of equations

where

.

Additionally, since this is a Natural Spline we see that and .

So in the end finding the cubic spline involves solving the system that I posted. All you have to do is plug things in.

I would say this belongs in the Advanced Applied Mathematics section.