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  "body": "\n## Summary\n\nNeville's algorithm computes the value at one point $x$ of the [[interpolation polynomial|SS1YMS2NPC]] through given data, without computing its coefficients.\nIt builds a triangular table.\nEach entry is the value at $x$ of the interpolation polynomial through a block of consecutive nodes.\nIt is computed from two entries for shorter blocks.\nIt costs $O(n^2)$ operations for $n + 1$ nodes.\nIt is often used for extrapolation, for example to the limit $h \\to 0$ in Richardson-type methods.\n\n## Explanations\n\n### L1 human\n\nThrough each single point there is a constant, through each pair a line, through each triple a parabola, and so on.\nNeville's idea is to get the value of the curve through a block of points by blending two shorter curves.\nOne goes through the block without its last point, the other through the block without its first point.\nThe blend is a weighted average whose weights depend on how far $x$ is from the two end nodes.\nFor the points $(0, 1)$ and $(1, 3)$, the line has value $1 \\cdot (1 - x) + 3 \\cdot x$, and at $x = 2$ this gives $5$.\nThe [[definition|KVZ7D84541]] gives the recursion.\n",
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