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  "body": "\n## Definition\n\nLet $K$ be a [[field|04C5ZBG1WT]], let $n \\ge 0$, and let $x_0, \\dots, x_n \\in K$ be pairwise distinct.\nLet $f$ be a function defined at these points with values in $K$.\nThe *Newton polynomial* of $f$ at $x_0, \\dots, x_n$ is\n$$N(x) = \\sum_{j=0}^{n} f[x_0, \\dots, x_j] \\prod_{k=0}^{j-1} (x - x_k),$$\nwhere $f[x_0, \\dots, x_j]$ are the [[divided differences|N5ZHV4HWDC]] of $f$ and the empty product for $j = 0$ is $1$.\n\n## Remarks\n\n$N$ has degree at most $n$ and $N(x_i) = f(x_i)$ for all $i$, so it is the [[interpolation polynomial|SS1YMS2NPC]] of the data, written in the Newton basis.\nThe first $j + 1$ terms form the interpolation polynomial at $x_0, \\dots, x_j$.\nIt can be evaluated with nested multiplication, as in [[Horner's method|FYBQ3Z52Z7]].\n\nExample: for $f(0) = 1$, $f(1) = 3$, $f(2) = 7$, the divided differences are $f[0] = 1$, $f[0, 1] = 2$, $f[1, 2] = 4$ and $f[0, 1, 2] = 1$.\nSo $N(x) = 1 + 2x + x(x - 1) = x^2 + x + 1$, and indeed $N(0) = 1$, $N(1) = 3$, $N(2) = 7$.\n\nNon-example: $1 + 2x + 2x(x - 1)$ has the Newton shape but its last coefficient is not $f[0, 1, 2]$; its value at $2$ is $9 \\ne 7$.\n\nFor equally spaced nodes $x_k = x_0 + kh$, the divided differences become forward differences divided by $k! \\, h^k$, which gives the [[Newton–Gregory forward formula|Q4YAST023H]].\n",
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