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  "body": "\n## Definition\n\nLet $n \\ge 0$, let $x_0, x_1, \\dots, x_n$ be pairwise distinct real numbers, and let $y_0, y_1, \\dots, y_n$ be real numbers.\nFor $j = 0, \\dots, n$, the *Lagrange basis polynomial* is\n$$\\ell_j(x) = \\prod_{\\substack{0 \\le m \\le n \\\\ m \\ne j}} \\frac{x - x_m}{x_j - x_m}.$$\nThe *Lagrange polynomial* of the data $(x_j, y_j)$ is\n$$L(x) = \\sum_{j=0}^{n} y_j \\, \\ell_j(x).$$\n\n## Remarks\n\nEach $\\ell_j$ has [[degree|0DPF6WRZW6]] $n$, with $\\ell_j(x_j) = 1$ and $\\ell_j(x_k) = 0$ for $k \\ne j$.\nSo $L$ lies in the [[polynomial ring|0Q9EZS90GY]] $\\R[x]$, has degree at most $n$, and satisfies $L(x_k) = y_k$ for all $k$.\nIt is the unique polynomial with these properties: the difference of two of them has degree at most $n$ and $n + 1$ roots, so it is $0$.\nThe same formula works over any field; this solves [[polynomial interpolation|SS1YMS2NPC]].\n\nExample: nodes $0, 1, 2$ with values $1, 3, 7$.\nHere $\\ell_0 = \\frac{(x - 1)(x - 2)}{2}$, $\\ell_1 = -x(x - 2)$ and $\\ell_2 = \\frac{x(x - 1)}{2}$.\nThen $L = \\ell_0 + 3 \\ell_1 + 7 \\ell_2 = x^2 + x + 1$, and indeed $L(0) = 1$, $L(1) = 3$, $L(2) = 7$.\n\nNon-example: if two nodes coincide, say $x_0 = x_1$, the factor $\\frac{x - x_1}{x_0 - x_1}$ in $\\ell_0$ has denominator $0$, so the formula is not defined.\n",
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