de Boor's algorithm

Remarks

The denominators are positive, since $t_i \le t_j < t_{j+1} \le t_{i+k-r}$ in the stated range, and $\alpha_{i,r} \in [0, 1]$, so each step is a convex combination. Correctness follows from the Cox–de Boor recursion: one step rewrites $\sum_i c_i B_{i,k}(x)$ as $\sum_i c_i^{[1]} B_{i,k-1}(x)$, and at order $1$ only $B_{j,1}(x) = 1$ survives. Example: $k = 2$, knots $t_0 = 0, t_1 = 1, t_2 = 2$, $x = 1/2$, so $j = 0$; then $\alpha_{0,1} = 1/2$ and $s(1/2) = (c_{-1} + c_0)/2$, which is the piecewise linear interpolant at the midpoint. Non-example: for $x$ outside $[t_j, t_{j+1})$ the same formulas with this $j$ do not give $s(x)$ in general, because other B-splines are active there. The cost is $O(k^2)$ operations per point.