{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $x_0 \\in \\R$, let $h \u003e 0$, and let $x_i = x_0 + i h$ for $i = 0, 1, \\dots, n$.\nLet $f$ be a real function defined at these points.\nDefine the forward differences by $\\Delta^0 f(x) = f(x)$ and $\\Delta^{k+1} f(x) = \\Delta^k f(x + h) - \\Delta^k f(x)$.\nLet $p$ be the interpolating polynomial of degree at most $n$ with $p(x_i) = f(x_i)$ for $i = 0, \\dots, n$.\nThen for every real $s$,\n$$p(x_0 + s h) = \\sum_{k=0}^{n} \\binom{s}{k} \\Delta^k f(x_0), \\qquad \\binom{s}{k} = \\frac{s (s - 1) \\cdots (s - k + 1)}{k!}.$$\n\n## Remarks\n\nProof idea: the [[Newton polynomial|9W77FPTM9E]] gives $p(x) = \\sum_{k=0}^{n} f[x_0, \\dots, x_k] \\prod_{j\u003ck} (x - x_j)$.\nFor equally spaced nodes, induction on $k$ shows that the [[divided difference|N5ZHV4HWDC]] is $f[x_0, \\dots, x_k] = \\Delta^k f(x_0) / (k!\\, h^k)$.\nWith $x = x_0 + s h$ one has $\\prod_{j\u003ck} (x - x_j) = h^k s (s - 1) \\cdots (s - k + 1)$, and the two formulas combine.\n\nExample: $f(x) = x^2$, $x_0 = 0$, $h = 1$, $n = 2$.\nThe values are $0, 1, 4$, so $\\Delta f(0) = 1$ and $\\Delta^2 f(0) = 2$.\nThe formula gives $p(s) = 0 + s + \\binom{s}{2} \\cdot 2 = s + s(s - 1) = s^2$, which is $f$ itself.\n\nThe formula needs equal spacing; for unequal nodes the general Newton form with divided differences must be used.\n\n## Explanations\n\n### L1 human\n\nGiven equally spaced values of a function, the formula builds the polynomial through them from a table of differences.\nThe first differences measure slope, the second differences measure curvature, and so on.\nThe weights are binomial coefficients in the position $s$, measured in steps of $h$.\nIt is the discrete analogue of the Taylor series, with differences in place of derivatives and $\\binom{s}{k}$ in place of $s^k / k!$.\n\n### L2 human\n\nThe formula is the basis of classical table interpolation and of many finite difference formulas.\nDifferentiating or integrating it term by term gives numerical differentiation rules and the Newton–Cotes formulas.\nThe error is that of [[polynomial interpolation|SS1YMS2NPC]]: if $f$ is $n + 1$ times differentiable, then $f(x_0 + s h) - p(x_0 + s h) = h^{n+1} \\binom{s}{n+1} f^{(n+1)}(\\xi)$ for some $\\xi$ in the smallest interval containing $x_0, x_n$ and $x_0 + s h$.\nThere is a matching backward formula, based at $x_n$ with backward differences.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "9W77FPTM9E"
    ],
    "kind": "statement",
    "msc": [
      "41A05",
      "65D05",
      "39A70"
    ],
    "name": "Newton–Gregory forward formula",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Newton_polynomial"
      }
    ],
    "tag": "Q4YAST023H",
    "uses": [
      "9W77FPTM9E",
      "N5ZHV4HWDC",
      "0GBZM0MT0F"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/Q4YAST023H/",
  "links": [
    {
      "field": "about",
      "target": "9W77FPTM9E"
    },
    {
      "field": "uses",
      "target": "9W77FPTM9E"
    },
    {
      "field": "uses",
      "target": "N5ZHV4HWDC"
    },
    {
      "field": "uses",
      "target": "0GBZM0MT0F"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/Q4YAST023H.md",
  "msc": [
    "41A05",
    "65D05",
    "39A70"
  ],
  "name": "Newton–Gregory forward formula",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Q4Y/Q4YAST023H.md",
  "status": null,
  "tag": "Q4YAST023H",
  "uri": "m2://Q4YAST023H",
  "url": "https://tamnd.github.io/math2/wiki/newton-gregory-forward-formula/",
  "writing": {
    "tag": "Q4YAST023H",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}