Say you have 20 points, , , to . And you know the value of a function f only in these points. Cubic splines are used to approximate the function f (from which you don't know the formula, or its value everywhere) between each intervals , , etc. The approximation is made by cubic polynomials on each interval.
You said :. This is not always true, but when it happens, we call it a natural cubic spline. It is made because if we don't use this condition, then we cannot find the cubic spline (because it has too much unknowns. In order to determine it, we add 2 conditions : its second derivative is equals to 0 at and at , or in our case .)I know the method involves the value of the second derivative being equated to 0 between points
The cubic spline must satisfy some other conditions, for example it must be continuous and its order must equals to 3 (in our case).
Maybe I can look if I can give you an example (just ask me if you would like to).