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  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]], let $n \\ge 0$, let $x_0, \\dots, x_n \\in K$ be pairwise distinct, and let $y_0, \\dots, y_n \\in K$.\nFor $j = 0, \\dots, n$, the Lagrange basis polynomial is\n$$\\ell_j(x) = \\prod_{k \\ne j} \\frac{x - x_k}{x_j - x_k} \\in K[x].$$\nThe Lagrange interpolation polynomial of the data is\n$$L(x) = \\sum_{j=0}^{n} y_j \\, \\ell_j(x).$$\n\n## Remarks\n\nEach $\\ell_j$ has degree $n$, with $\\ell_j(x_j) = 1$ and $\\ell_j(x_k) = 0$ for $k \\ne j$.\nSo $L$ has degree at most $n$ and $L(x_i) = y_i$ for all $i$: it is the unique [[interpolation polynomial|SS1YMS2NPC]] of the data.\nExample: for the data $(0, 1)$, $(1, 3)$, $(2, 7)$, the basis polynomials are $\\ell_0 = \\frac{(x-1)(x-2)}{2}$, $\\ell_1 = -x(x-2)$ and $\\ell_2 = \\frac{x(x-1)}{2}$.\nThen $L = \\ell_0 + 3 \\ell_1 + 7 \\ell_2 = x^2 + x + 1$, and indeed $L(0) = 1$, $L(1) = 3$, $L(2) = 7$.\nNon-example: if two nodes coincide, $x_j = x_k$ with $j \\ne k$, the denominator $x_j - x_k$ is $0$ and the formula has no meaning.\n\n## Explanations\n\n### L1 human\n\nEach basis polynomial is a switch that is on at one node and off at all others.\nWeighting the switches by the wanted values gives a polynomial through all the points.\n",
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