This is the third of a series of article about spline approximation. If you have not done so, you should start reading
So a function which is supposed to approximate a given set of points as a linear combination
of functions , as described in the previous article with
and with chosen such that for all
Now up to four s can overlap in one sub-interval. Assuming the subinterval is
and the arithmetic mean of the borders is
This interval is touched by the base functions , so the function can be written for as
Now we have
and substitute that:
This will be used to actually calculate g in programs.
Disclaimer: I am not an expert in numerical analysis. While I believe that the approach that this article comes up with is sound and useful, I do believe that an expert of numerical analysis could still improve the accuracy of the calculations.
How to actually program it will be covered in the upcoming article „Spline Approximation (Cookbook)“. The link will be added, when it is available.