Custom Cubic Interpolation

This example goes over determining whether we can express points with two piecewise cubic functions.

Problem

Given can we define with 2 piecewise cubic functions?
Such that interpolates data points & is continuous. Write a linear system to find .

Solution:

First write out the linear system:
Now we can write it as a compact system:
By looking at the system we can see that there are 7 equations and 8 unknowns so there will be infinitely many solutions and we thus need one more constraint to have a well defined interpolation.