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  "body": "\n## Definition\n\nLet $k \\ge 1$ be an order and let $(t_i)$ be a nondecreasing knot sequence.\nLet $s = \\sum_i c_i B_{i,k}$, where $B_{i,k}$ are the [[B-splines|62AFX32JGT]] of order $k$ and the $c_i$ are real numbers or points of $\\R^d$.\nLet $x$ be a point with $t_j \\le x \u003c t_{j+1}$.\n*De Boor's algorithm* computes $s(x)$ as follows.\n\n1. Set $c_i^{[0]} = c_i$ for $i = j - k + 1, \\dots, j$.\n2. For $r = 1, \\dots, k - 1$ and $i = j - k + 1 + r, \\dots, j$, set\n$$\\alpha_{i,r} = \\frac{x - t_i}{t_{i+k-r} - t_i}, \\qquad c_i^{[r]} = (1 - \\alpha_{i,r})\\, c_{i-1}^{[r-1]} + \\alpha_{i,r}\\, c_i^{[r-1]}.$$\n3. Return $c_j^{[k-1]}$, which equals $s(x)$.\n\n## Remarks\n\nThe denominators are positive, since $t_i \\le t_j \u003c t_{j+1} \\le t_{i+k-r}$ in the stated range, and $\\alpha_{i,r} \\in [0, 1]$, so each step is a [[convex combination|21SXYJYC59]].\nCorrectness follows from the [[Cox–de Boor recursion|WVD069C7QF]]: 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.\nExample: $k = 2$, 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.\nNon-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.\nThe cost is $O(k^2)$ operations per point.\n",
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