Clamped cubic spline interpolant

Let $a=x_0<x_1<\cdots <x_m=b$, with $m\ge 1$, let $y_0,\dots ,y_m\in \mathbb{R}$, and prescribe endpoint slopes ${\sigma }_a,{\sigma }_b\in \mathbb{R}$. A clamped cubic spline interpolant for these data is a function $s\in C^2([a,b])$ whose restriction to every $[x_i,x_{i+1}]$ is a polynomial of degree at most three, which satisfies $s(x_i)=y_i$ for every $i$, and which satisfies $s^{\mathrm{\prime }}(a)={\sigma }_a$ and $s^{\mathrm{\prime }}(b)={\sigma }_b$.

Here $C^2$ means that the function and its first two derivatives are continuous on $[a,b]$, with one-sided derivatives at the endpoints. This uses the piecewise-cubic convention in Natural cubic splines and replaces its natural endpoint-curvature conditions by prescribed endpoint slopes. It does not alter the natural cubic interpolant or its system.