The case n = 2 for parabolically terminated cubic splines is not covered by Theorem 3.8. Discuss existence and uniqueness for the cubic spline in this case.
Assume that n ≥ 2. Then, for a set of data points (x1, y1),… , (, ) and for any one of the end conditions given by Properties 4a–4c, there is a unique cubic spline satisfying the end conditions and fitting the points. The same is true assuming that n ≥ 3 for Property 4d and n ≥ 4 for Property 4e.
Parabolically terminated cubic spline. The first and last parts of the spline, S1 and , are forced to be at most degree 2, by specifying that d1 = 0 = . Equivalently, according to (3.22), we can require that c1 =c2 and = . The equations form the two tableau rows