Past exam of the options I course of the University of Oxford / 2008 / P / P2 / Solution

A natural cubic spline is piecewise cubic, interpolates the data, has continuous first and second derivatives at the knot, and has zero second derivative at the outer endpoints. Its nodal second derivatives here satisfy $M_0=M_2=0$ and the usual matching equationso $M_1=6$. The resulting pieces areDirectly, the values are $0,0,4$, both first derivatives at one equal two, both second derivatives there equal six, and $S^{\prime \prime }(0)=S^{\prime \prime }(2)=0$. Thus all spline conditions hold.

The second data set adds the values of the affine function $x$ to the first. Adding $x$ to the spline preserves derivative matching and the zero endpoint second derivatives, so uniqueness of the natural interpolant gives the new spline $S(x)+x$. Its first piece is $x^3$, and its second is $1+3t+3t^2-t^3$. This is the affine covariance of natural cubic spline interpolation.