Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 43 / 5 / b / Solution

A natural cubic spline interpolant $s$ satisfies $s(x_i)=y_i$, is twice continuously differentiable, and is a polynomial of degree at most three on each interval $[x_i,x_{i+1}]$. Its natural boundary conditions areIf defined outside the data interval, it continues linearly on both exterior intervals. Thus function values and the first two derivatives match at the knots, but the third derivative may jump.

For an explicit construction, let $h_i=x_{i+1}-x_i>0$ and $M_i=s^{\prime \prime }(x_i)$. Set $M_1=M_n=0$ and solveOn $[x_i,x_{i+1}]$, putThis formula interpolates the values and matches second derivatives; the tridiagonal equations enforce first-derivative matching. The homogeneous system has only the zero solution: in a row with maximal nonzero $|M_i|$, its diagonal coefficient strictly exceeds the sum of its neighbour coefficients, making cancellation impossible. Thus the natural interpolant exists uniquely.